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Cite this article: Gou K, Chen Z (2015) Inverse Sturm-Liouville Problems and their Biomedical Engineering Applications. JSM Math Stat 2(1): 1008.

*Corresponding authorKun Gou, Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA, Tel: 951-468-5861; Email:

Submitted: 17 December 2014

Accepted: 17 April 2015

Published: 21 April 2015

Copyright© 2015 Gou et al.

OPEN ACCESS

Keywords•Inverse spectral problem•Sturm-Liouville problem•Intravascular ultrasound•Atherosclerosis

Review Article

Inverse Sturm-Liouville Problems and their Biomedical Engineering ApplicationsKun Gou1* and Zi Chen2

1Department of Mechanical Engineering, Michigan State University, USA2Thayer School of Engineering, Dartmouth College, USA

Abstract

Sturm-Liouville Problems (SLPs) arise in many applied mathematics and physics problems, such as vibrations of strings, plasma dynamics, and Schrodinger equations in quantum mechanics, and have been extensively studied. Inverse SLPs have also been broadly investigated, but many important issues remain to be addressed. Here, we provide an overview of SLPs and inverse SLPs, and emphasize the emerging applications of inverse SLPs in biomedical engineering. By doing so, we hope to garner due attention about these fundamental problems from the mathematics, physics, and engineering communities and foster further studies of the open problems that are not only important in mathematical sciences but also have broad engineering applications.

ABBREVIATIONSSLP: Sturm-Liouville Problem; IVUS: Intravascular Ultrasound

INTRODUCTIONSturm-Liouville equations date back to more than 150 years

ago and have been well studied. The inverse SLPs are more challenging because of the variety of formulations that are difficult to obtain and justify. In this article, we review inverse spectral SLPs and their applications in biomedical technology. A short summary of the theoretical and numerical development of inverse SLPs is given. A novel formulation of inverse SLPs by virtue of dual SLP operators is presented as an example of more complicated inverse SLPs. The proof for local existence is given as an initial exploration for this dual-operator problem. For a general inverse SLP, a numerical formulation and techniques for approximation of the unknown functions are demonstrated. Biomedical application of inverse SLPs is also illustrated, in which the intravascular ultrasound technique is novelly employed to determine the natural frequencies of the arterial wall. Via an asymptotic approach, an inverse SLP is employed to recover the arterial wall material parameters for potentially more accurate discrimination between stable and unstable atherosclerotic plaques.

Direct sturm-liouville problems

The classical SLPs are named after two French mathematicians, Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), in memory of their contributions to the development of the problems. SLPs arise in many physical problems [1]. One

classical example is the planar vibrations of strings, such as those on the violin [2]. The transverse displacement satisfies a hyperbolic differential equation where time and position are the independent variables. SLPs are obtained by solving the equation using the method of separation of variables. Another famous example is the one-dimensional Schrödinger wave equation in quantum mechanics, which also adopts the form of SLPs [3]. In mathematical analysis, the Bessel equation [4] and the Legendre equation [5] can both be studied by the theory of SLPs. In fact, SLPs establish the relationship between the physical properties and the eigenfrequencies and are foundations for more profound analysis between these two sets of quantities.

A classical regular Sturm-Liouville problem in its typical form [6] is a real second-order linear differential equation:

( ( ) ( )) ( ) ( )= ( ) ( ) ( , ),p x u x q x u x W x u x for x a bλ′ ′− + ∈ (1.1)

with boundary conditions of the form

1 1( ) ( )= 0,u a u aα β′ + (1.2)

2 2( ) ( )= 0,u b u bα β′ + (1.3)

where ( )p x , ( )q x , and ( )W x are real-valued piece-wise continuous functions with at most finite jumps of discontinuities over the finite interval [ , ]a b , satisfying ( )> 0p x and ( )> 0W x for any [ , ]x a b∈ . Here, λ is the eigenvalue and the corresponding solution ( )u x is called the eigenfunction. The four parameters in the boundary conditions 1α , 1β , 2α and 2β satisfy 2 2

1 1 0α β+ ≠ and 2 2

2 2 0α β+ ≠ . If 1 2= = 0β β , the boundary conditions are called Dirichlet boundary conditions, while if 1 2= = 0α α , they are called Neumann boundary conditions. Other cases are called boundary

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conditions of the mixed type. For a regular SLP, the eigenvalues are simple. Namely, each eigenvalue generates only one linearly independent eigenfunction. All the corresponding eigenvalues can be ordered to be a sequence which tends to positive infinity such that 0 1 2 3< < < <λ λ λ λ ⋅⋅⋅ .

In practice, some of the engineering or applied problems produce second-order differential equations of the form

( ) ( ) ( ) = ( ) ,F x u H x u G x u E x uλ′′ ′+ + (1.4)

other than a SLP in (1.1) directly [7,8]. Then (1.4) can be converted into the form (1.1) by virtue of the relations given in Appendix A.

Under more constraints for the coefficients ( )p x , ( )q x and ( )W x , SLPs in the form (1.1-1.3) can be further transformed by

the Liouville transformation [9] into the following

( ) ( ) ( )= ( ) (0,1),u x Q x u x u x xλ′′− + ∈ (1.5)

(0) (0)= 0,u hu′ − (1.6)

(1) (1)= 0,u Hu′ + (1.7)

Where h and H are two constants and can also be ∞ . SLPs in the form of (1.5-1.7) are the most commonly used ones due to their simplicity. However, for some applied problems, the coefficients ( )p x , ( )q x and ( )W x obtained from a second-order differential equation (1.4) may be highly complicated [8,10]. The potential ( )Q x generated by the Liouville transformation is even more complicated and therefore causes substantial difficulty for further processing. Under this situation, direct study on (1.1) should be considered.

Inverse sturm-liouville problems

A direct SLP allows the calculation of eigenvalues and eigenfunctions through the SLP differential equation and boundary conditions. Conversely, the inverse SLP allows the derivation of unknown functions (including unknown parameters) involved in the coefficients from the eigenvalue data λ . Inverse SLPs have broad applications. For example, for a vibrating string of known vibrating frequencies, one can derive the density of the string using an inverse SLP [11,12]. In addition, one can also describe the density profile of the planet earth from information of the vibrating modes of the earth shell [13]. In quantum mechanics, an inverse SLP can generate the potential

( )Q x from the energy levels λ of the system [3].

Inverse recovery of ( )Q x for the SLP in (1.5-1.7) has been broadly investigated. Some results, however, are beyond intuition. For example, it has been shown that a single set of spectra, =1{ }n nλ ∞ , alone does not offer sufficient information to recover a unique ( )Q x [14]. Instead, Gel’fand and Levitan [15] showed that ( )Q x could be uniquely found from =

=1{ }nn nλ ∞ and

a set of constants 2 ==1{ =|| ||}n

n n nuη ∞ for the eigenfunction nu of the corresponding eigenvalue nλ . If Eqs. (1.6) and (1.7) give Dirichelet boundary conditions under which = =h H ∞ , then

=1{ }n nλ ∞ and =1{ = log(| (1)/ (0)|)}n n n nu uκ ∞′ ′ suffice to determine ( )Q x uniquely. Notice that nκ requires less information from nu than nη , which is a balance between different boundary

conditions. If, instead, only the boundary condition at = 1x is the Dirichlet type, we can define =1{ = log(| (1)/ (0)|)}n n n nu uκ ∞′ to uniquely recover ( )Q x in conjunction with =1{ }n nλ ∞ . For all the

special cases, see [16,17] for more detail.

The boundary conditions play a more important role in the formulation of inverse SLPs. Marchenko [18] showed that with the same equation (1.5), if we obtain another set of eigenvalues, =1{ }n nγ ∞ , from another pair of boundary conditions,

ˆ(0) (0)= 0, (1) (1)= 0u hu u Hu′ ′− + , where H H≠ , then the two sets of spectra =1{ }n nλ ∞ and =1{ }n nγ ∞ uniquely determine the potential

( )Q x , h, H and H . The proof herein forms a foundation for a reconstruction algorithm for ( )Q x [19]. If some eigenvalues of =1{ }n nγ ∞ can not be obtained, then Hochstadt [20] pointed out that the Fourier expansion of ( )u x will miss the related modes corresponding to the missing eigenvalues. A numerical reconstruction of ( )Q x by using truncated Fourier series is based on this result [21].

If ( )Q x is symmetric about the axis = 1/2x , i.e., (1 )= ( )Q x Q x− , and =h H , then =1{ }n nλ ∞ is enough to uniquely determine ( )Q x . If

( )Q x is not symmetric but known on the interval [1/2,1] , then one set of spectra is enough to recover the other half of ( )Q x over [0,1/2] [22]. Gou and Sun [23] also designed a numerical procedure for solving one kind of Goursat problem arising from the inverse problem with half of the potential known in advance.

From the short review of the various inverse SLPs above, we can imagine that many other inverse SLPs can be established based on the variation of the boundary conditions, information of the potential or spectral data and so on. Unfortunately, many of these more profound issues remain unexplored. In Section 2, one inverse SLP from dual operators with identical boundary conditions is presented. Analysis for local existence of the solution is supplied. While inverse SLPs in the form of (1.5-1.7) attract much attention, more complicated inverse SLPs in the form of (1.1-1.3) is also significant. More specifically, the coefficients ( )p x , ( )q x and ( )W x , and the boundary constants

1α , 1β , 2α and 2β all involve unknown functions. The numerical formulation and approximation for the unknowns are significant for a reliable and efficient recovery. Section 3 gives techniques for resolving these issues .

There are also newly emerging advanced applications of inverse SLPs on the biomedical sciences. For instance, by using the intravascular ultrasound technique, one can find the natural frequencies of blood vessels. Via formulating possible inverse spectral problems, we can recover the wall tissue’s material and mechanical properties. This technique can facilitate detection of some diseases uneasily observed by conventional techniques. Section 4 illustrates this usage in more detail.

DUAL OPERATOR INVERSE SLPsIn this section, inverse SLPs with two equations (dual

operators) are studied. We define two SLPs with Dirichilet boundary conditions:

SLP1:

( ) = , (0,1)u Q x u u xλ′′− + ∈ (2.1)

(0)= 0,u (2.2)

(1)= 0;u (2.3)

SLP2:

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( ( ) ( )) = , (0,1)v Q x p x v v xµ′′− + + ∈ (2.4)

(0)= 0,v (2.5)

(1)= 0,v (2.6)

where ( )p x , =1{ }j jλ ∞ and =1{ }j jµ ∞ are given. ( )Q x is reconstructed from combination of the two SLPs.

Lemma 2.1 Suppose ( )p x is symmetric about = 1/2x , i.e. ( )= (1 )p x p x− , and ( )Q x is the solution of SLP1 and SLP2. Then (1 )Q x− is also a solution.

Proof. For any ( )xΘ , ( ) =u x u uλ′′− +Θ and (1 ) =u x u uλ′′− +Θ − with Dirichlet boundary conditions have the same spectral data. Since, ( )= (1 )p x p x− , (1 ) (1 )= (1 ) ( )Q x p x Q x p x− + − − + , so (1 )Q x− is also a solution for SL2. Obviously, (1 )Q x− is a solution for SLP1. This proves the lemma.

Without loss of generality, we ignore the cases under which ( )= (1 )p x p x− , which only give unique solutions up to symmetry

about = 1/2x .

Because of the Dirichlet boudnary conditions, SLP1 gives 2 2

1= , 1,j jj s for jλ π λ+ + ≥ (2.7)

where 1

1 0= ( )s Q x dx R∈∫ and 2

1 2 3=( , , , ) lλ λ λ λ ∈

. Similarly, SLP2 gives

2 22= , 1,j jj s for jµ π µ+ + ≥ (2.8)

where 1 1

2 0 0= ( ) ( )s Q x dx p x dx R+ ∈∫ ∫ and 2

1 2 3=( , , , ) lµ µ µ µ ∈ . For

convenience, we assume that 1 = 0s and 2 = 0s . Then 2 2=j jjλ π λ+ and 2 2=j jjµ π µ+ .

Lemma 2.2 Let 0 1

= ( )minminx

p p x≤ ≤

and 0 1

= ( )maxmaxx

p p x≤ ≤

. If for some

k N∈ , [ , ]k k min maxp pµ λ− ∉ , then there is no solution for ( )Q x .

Proof. Suppose such ( )Q x exists. By Theorem 2.8 on [35], SLP1 and SLP2 yield

, .min k k maxp p foreach k Nµ λ≤ − ≤ ∈

, .min k k maxp p foreach k Nµ λ≤ − ≤ ∈

(2.9)

This gives a contradiction.

If =min maxp p , i.e., ( )p x c≡ for some c R∈ , then by (2.9),

k k cµ λ− ≡ for each k . The eigenvalues in SLP2 are a pure translation of eigenvalues in SLP1 by c. As such, infinitely many solutions can be obtained. Therefore, ( )p x is assumed not to be a constant.

To illustrate the connection of the spectra of SLP1 and SLP2 with Q, the spectra are denoted by =1{ ( )}j jQλ ∞ and =1{ ( )}j jQµ ∞ respectively. Define space

1 1

2 2

= { : =( ( ), ( ),( ), ( ), ... ( ), ( ), ...)},n n

S Q QQ Q Q Qρ ρ λ µ

λ µ λ µ (2.10)

and construct a function from 2L to S by

1 1

2 2

( ):=( ( ), ( ),( ), ( ), ... ( ), ( ), ...).n n

Q Q QQ Q Q Q

ρ λ µλ µ λ µ

(2.11)

The space S is not an open space of Banach space. However, due to the one-to-one mapping between S and

{ ( ) ( ) ( )(( ) ( ) ( ) )}

1 1 2

2

: , , ,

, ... , , ...n n

S Q Q Q s

Q Q Q

ρ ρ λ µ λ

µ λ µ

= =

and 2S l⊆ , S is an open subspace of Banach space. We can thus work on S

and treat it as a standard coordinate system.

Lemma 2.3 ( )Qρ is a real analytic map from 2L to S with its derivative out Q given by

1 12 2

0 0( )=( ( 1) , ( 1) , 1),Q n nd f u fdx v fdx nρ − − ≥∫ ∫ (2.12)

where 2( )f x L∈ , and ( )nu x and ( )nv x are the normalized eigenfunctions of the nth mode of SLP1 and SLP2 respectively, i.e.,

1 12 2

0 0= = 1n nu dx v dx∫ ∫ .

Proof. By Theorem 3.1 in [24], for SLP1 there exists neighborhood 1U of Q such that all eigenvalues =1{ ( )}j jQλ ∞ extend analytically to it. Similarly for SLP2, one can find a neighborhood U2 of Q such that all eigenvalues =1{ ( )}j jQµ ∞ extend analytically to it. Let 1 2=U U U∩ . Then both =1{ ( )}j jQλ ∞ and =1{ ( )}j jQµ ∞ extend analytically to U. By

12 2

0

1( )= (2 ) ( )j Q j Qcos jx dx On

λ π π− +∫ (2.13)

and 12 2

0

1( )= ( ) (2 ) ( ),j Q j Q p cos jx dx On

µ π π− + +∫ (2.14)

we know the extension of ( )Qρ is bounded on U and the coefficients of each 2l sequence in S all are analytic. Therefore, the map ( )Qρ is an analytic extension from U to 2l .

By Theorem 2.3 in [24], the gradient of ( )n Qλ with respect to Q is 2 1nu − . The chain rule for Fréchet derivative produces

2( ) = 1.nn

Q vQ

µ∂−

∂

(2.15)

This establishes the result in (2.12).

Theorem 2.4 (Local existence) for any 2( )Q x L∈ and spectra

=1{ ( )}j jQλ ∞ and =1{ ( )}j jQµ ∞ from SLP1 and SLP2 respectively, there

exists a neighborhood V of 2 2=1 =1({ ( )} ,{ ( )} ) ( , )j j j jQ Q l lλ µ∞ ∞ ⊂ such

that there exists ( )q x near ( )Q x generating the perturbed pairs of eigenvalues by SLP1 and SLP2 of each Vν ∈ .

Proof. The subspace (denoted as 1H ) of 2L perpendicular to the space spanned by 1 and 2{ 1: 1}nu n− ≥ is the space spanned

by 2( ){ : 1}ndu x n

dx≥ (Theorem 3.9 in [24]); Similarly, the subspace

(denoted as 2H ) of 2L perpendicular to the space spanned by 1

and 2{ 1: 1}nv n− ≥ is the space spanned by 2( ){ : 1}ndv x ndx

≥ . Let

1 2=H H H∩ . Then the kernel of the operator ( )Qd fρ is H. Let K be the orthogonal complement of H in 2L .

We restrict ( )Qρ on K and denote it by ( )K Qρ . ( ) 0Q Kd fρ ≠ for f K∈ , so by the Implicit Function Theorem, ( )K Qρ is a real analytic isomorphism between a neighborhood Θ of ( )Q x and a neighborhood V of 2

=1{ ( ), ( )}j j jQ Q lλ µ ∞ ⊂

in S. This proves the theorem.

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FORMULATION AND APPROXIMATIONMany well-known numerical techniques for reconstruction

of inverse SLPs have been accomplished [25,26,19,27]. None of these is dominant over the others. Some of the techniques are more accurate but burdensome to realize, while others are easier to apply but less accurate. Various other techniques can still be designed. This section presents different reconstruction algorithms.

Formulation

We consider the SLP (1.1-1.3) with the domain [0,1] . The prüfer transformation transforms the second-order SLPs (1.1) to a pair of first-order nonlinear equations by introducing two variables called prüfer amplitude and prüfer phase denoted by

( )xρ and ( )xθ respectively [28]. The relations between ( )u x , ( )xρ and ( )xθ are

( ) ( ) ( )( )( ) ( ) ( ) ( )( )

sin ,

' cos .

u x x x

p x u x x x

ρ θ

ρ θ

=

= (3.1)

Plugging (3.1) into (1.1) produces

2 21( )= ( ( )) ( ( ) ( )) ( ( )),( )

x cos x W x q x sin xp x

θ θ λ θ′ + − (3.2)

( ) 1 1= ( ( ) ( )) (2 ( )).( ) 2 ( )

x W x q x sin xx p x

ρ λ θρ′

− + (3.3)

By (1.2) and (1.3), the boundary conditions for ( )xθ are correspondingly

1

1

(0)= ( ) (0) [0, ),(0)

arctan forpα

θ θ πβ

− ∈ (3.4)

2

2

(1)= ( ) ( 1)(1)

(1) ( 1) (0, ], = 0,1,2

arctan np

for n n

αθ π

βθ π π

− + +

− + ∈

(3.5)

The extra boundary condition (3.5) for ( )xθ provides valuable information. In the Sleign2 algorithm [29], it is used to compute the eigenvalues λ . It is also employed to recover the unknown function ( )xτ for the inverse SLPs.

Suppose the unknown function ( )xτ is incorporated in ( )p x , ( )q x or ( )W x (the coefficients 1α , 1β , 2α and 2β in the

boundary conditions may also depend on data of ( )xτ . An implicit function is defined for ( )xτ and each eigenvalue =1= { }j n nλ λ ∞∈Λ from (27) by

2=1

2

( , )= ( )| ( ) ( 1) ,(1)

jj xd x arctan j

pβ

τ λ θ πα

+ − +

(3.6)

where ( )j xθ is the solution of the initial value problem (24) and (26) by replacing λ by jλ . If τ is the real solution, then

( , )= 0.jd τ λ (3.7)

Recovering ( )xτ requires solving all these nonlinear equations formed by (3.7).

Approximation for the unknown

Approximation by the Fourier series: Let ( )xτ for [0,1]x∈ have a Fourier expansion as

0

=1( )= ( cos(2 ) sin(2 )).

2 n nn

ax a n x b n xτ π π

∞

+ +∑ (3.8)

To completely recover ( )xτ , all the coefficients need to be determined. Therefore, infinitely many equations in the form of (3.6) are required. Practically, one can at best obtain a finite number of eigenvalues in the set Λ to reconstruct an approximation for ( )xτ . Suppose we have δ consecutive eigenvalues denoted as

=1= { } .n nδλΛ (3.9)

The number of coefficients determined in (30) will equal the number of eigenvalues in the set Λ . We denote 0a , 1a , 1b , 2a ,

2b ... (totally δ values) as 1τ , 2τ , 3τ , 4τ , 5τ ... δτ . In a minor abuse of notation, we denote τ by

1 2=( , , ... ).δτ τ τ τ (3.10)

Therefore, the inverse SLP amounts to recovering τ in the form of (3.10).

Approximation by interpolation: The set of finite eigenvalues Λ is as in (3.9). The domain [0,1] is uniformly partitioned into 1δ − subintervals, where the nodes are denoted as 1x , 2 ,x xδ . The value of τ at ix , 1 i δ≤ ≤ , is denoted by iτ . The δ points ( , )i ix τ are interpolated by cubic splines

1

( )= ( ),,1 1,

i

i i

x P xfor x x x iτ

δ+≤ ≤ ≤ ≤ − (3.11)

where each ( )iP x is a cubic polynomial satisfying

1 1: ( )= , ( )= ,i i i i i icontinuity P x P xτ τ+ + (3.12)

1 1 1: ( )= ( ),i i i ifirst orderdifferentiability P x P x+ + +′ ′− (3.13)

1 1 1: ( )= ( ),i i i isecond orderdifferentiability P x P x+ + +′′ ′′− (3.14)

and the first and second derivatives at the two ends 1 = 0x and = 1xδ are set according to one’s preference for the corresponding

polynomials. Consequently, recovering τ amounts to estimating iτ for 1 i δ≤ ≤ , and the function approximated is given by (3.11).

Remark: there are other interpolation approaches that may also be used, e.g., the nearest-neighbor interpolation, the linear spline interpolation, the piecewise cubic spline interpolation, the Hermite interpolation and so on. Each kind is suitable for a different category of functions. For instance, the nearest-neighbor interpolation is the best choice for approximating piecewise constant function, and piecewise cubic spline interpolation gives an approximation without the second-order differentiability. We mainly make use of cubic spline interpolation for recovering the unknown τ under the assumption that it is typically a normal smooth function.

BIOMEDICAL APPLICATIONS

Background

It is well-known that cardiovascular diseases are a great threat to the human lives. For example, atherosclerosis is the principal cause of heart attack, stroke, and gangrene of the extremities, and accounts for approximately 50% of mortality in many developed countries [30]. It is a process in which cholesterol and other fatty deposits accumulate to form plaques along the inner wall of arteries that limits or sometimes even blocks blood flow. During the development of an atherosclerotic plaque, a softer and fatty

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core forms initially attached to the inner arterial wall, and then a calcified and fibrous layer forms over it towards the lumen of the arterial wall. According to the relative thickness of the two parts, two kinds of plaques can be formed. If a thick fibrous outer part stabilizes the core of build-up reducing the possibility of a spontaneous rupture, such plaque is called stable or safe, and no treatment is urgently needed for such condition. Otherwise, if the inner core is covered by a relatively thin outer layer, the fatty plaque is susceptible to fracture and thus called unstable or vulnerable [31,32]. After the rupture of the outer part, the fatty inner core can be transported into the bloodstream producing thrombosis and blocking the bloodstream through the vessel. The timely identification of unstable atherosclerotic plaques is a critical difficulty posed to all the cardiovascular researchers. The realization largely depends on accurate discrimination between the plaques with thin fibrous caps and thick lipid cores and the ones with thick fibrous caps and thin lipid layers.

To discriminate fibrous and fatty plaque tissues effectively, various Intravascular Ultrasound (IVUS) approaches for quantitatively characterizing tissues have been invented. IVUS imaging is an invasive procedure performed through cardiac catheterization that produces detailed images of the interior walls of the artery to see blood vessels from the inside [33,34]. IVUS imaging allows one to detect an obstruction of the lumen of the artery by careful examination of the imaging. On the distal tip of the catheter is an extremely miniaturized transducer that sends small amplitude, high frequency ultrasound waves (usually in the 15-40 MHz range) to the arterial wall and records the resulting echo. The echo is then transmitted back to a computerized ultrasound processor that generates detailed images of the interior arterial wall. By this approach, one can see from inside out a thin section of the blood vessel surrounding the catheter tip. Figure 1 displays a schematic view of an IVUS catheter inside an artery and a sample image it produces.

This technique is often used for diagnosis of atherosclerosis. However, to discriminate between safe and unsafe atherosclerotic plaques accurately for proper treatment, the results from IVUS are unsatisfactory [35-37]. They have low spatial resolution and give poor images of deeper plaques. Capability of IVUS to discriminate between fatty and fibrous plaque tissues lies in

the range of 39-52%. Achieving satisfactory results by means of traditional techniques like IVUS imaging is very difficult due to the complex nature of the tissue characterization and the incapable imaging techniques.

Based on the IVUS operational principle, we consider a novel use of the IVUS technique for a more accurate discrimination of plaques [7]. Under the IVUS interrogation, the ultrasound wave propagates through the vessel wall and imposes a periodic force on it. This causes a nanoscale small amplitude and high frequency vibration of the arterial wall. One can control the frequency of the ultrasound wave to produce arterial wall resonances. Several natural frequencies of the wall tissue can thus be obtained. These eigenfrequencies contain crucial information of plaque tissues, which can be explored to distinguish plaques better [7]. These natural eigenfrequencies are used to calculate the mechanical and geometrical parameters and other important parameters beneficial for more accurate discrimination of the stable and unstable plaques. Gou et al. recovered the shear modulus and residual stress as an initial set up of inverse problem for realization of such purpose on healthy arteries [8,10].

Equilibrium equations

As the IVUS is interrogating inside the artery [35,33], it produces small-amplitude high-frequency time-harmonic vibrations superimposed on the quasistatic deformation of the pre-stressed artery resulting from pulsatile blood flow. In the modeling framework, two kinds of boundary value problems are established according to two categories of deformations respectively via asymptotic analysis. The equilibrium equations under the static deformation = 0,Div bσ + are utilized for these derivations, where Div is the divergence operator, σ is the Cauchy stress tensor, and b is the body force (ignored for the light weighted soft tissue). Under the axisymmetric deformation, the unknown response functions only depend on the radius R of the cylinder.

Static deformations: The inner wall subjected to blood pressures produces a class of axisymmetric static deformations of the whole geometrical body. The outer boundary of the whole cylinder is assumed to be stress free (as in an in vitro experiment when an artery is excised away from its outer tissue and a

Figure 1a A schematic display of IVUS catheter operating inside an artery emitting ultrasound wave.

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pressure is imposed on its inner wall). The inner boundary is assumed to be subjected only to constant blood pressures iπ (i can be varied to indicate different quantities of blood pressures). The boundary value problem for the radial displacement function

0 ( )i

r Rπ by virtue of the equilibrium equation is denoted as

0 0 0( ( ), ( ), ' ( )) = 0,i i i

F r R r R r Rπ π π′ ′

Where R is the radius of the cylindrical wall and F is a known function derived from the analysis. The radius-dependent unknown mechanical or material parameters are implicitly involved in the equation.

Vibration caused deformations: Under the IVUS implementation, the inner wall is subjected to two pressures: the pulsatile blood pressure and the small amplitude, high frequency ultrasound wave generated pressure. These two can be connected to form a time harmonic blood pressure as = (1 ),i t

i e ωπ π ε+ where ε is an infinitesimally small quantity and ω is the frequency of IVUS wave. The radial displacement under such pressure is expanded as 0 1( , ) = ( ) ( ) ( ),i t

i i ir R t r R r R e oωπ π πε ε+ +

where 0 i

r π is the radial displacement caused by the static deformation, and

1 ( )i

r Rπ is the radial displacement caused by the perturbation from vibration. t is time but here does not enter into the final equations as we focus on the static situation. After asymptotic expansions, the vibration caused deformation produces another boundary value problem for 1 ( )

ir Rπ in a SLP form

1 1 1( ( )( ( )) ) ( ) ( ) = ( ) ( ),i i i i i i

p R r R q R r R W R r Rπ π π π π πλ′ ′− +

0 1 1 1

0 1 1 1

( ) ( ) = 0 = ,

( ) ' ( ) = 0 = ,Ii i i i

Oi i i i

a r R a r R for R R

b r R b r R for R Rπ π π π

π π π π

′+

+

where( )

ip Rπ , ( )

iq Rπ and ( )

iW Rπ are abbreviations of 0( , )

i ip R rπ π , 0( , )

i iq R rπ π

and 0( , )i i

W R rπ π respectively, IR and OR are the radii of the most

inner and most outer boundaries of the cylinder respectively on the undeformed deformation, and the eigenvalues λ are from the natural frequencies of the cylindrical wall gained by causing resonance of the wall vibration under IVUS interrogation. The above boundary value problem can be viewed as a free response function of the radial displacement.

Some remarks are made: (1) 0 ir π is a function of the

unknown functions and blood pressure iπ ; (2) i

pπ , i

qπ , i

Wπ are all functions of 0 i

r π , which means that they both implicitly and explicitly depend on the unknowns; (3) 0 i

a π , 1 ia π , 0 i

b π and 1 ib π

display the same involvement of the unknowns.

Solution procedure

The IVUS technique supposedly only obtains several lower-mode eigenfrequencies of the vessel wall for each blood pressure with relative accuracy. An approximation of an unknown function

( )f R is herein recovered by this limited information. Instead of recovering the whole function directly, we recover a vector of nodal values first. Then the vector is interpolated by (piecewise) cubic splines given in Section 3.2.2 to approximate the function.

The multidimensional secant method may be employed to obtain the solution (Appendix B). One may also use the technique given in Section 3.1. The following equations are then established

1

0

( , ) = ( ) ( ) ( 1) = 0.( )

iOi i

Oi i

bD f R arctan n

p R bπ

π ππ π

λ θ π+ − +

Different from the Multi-dimensional secant method, one blood pressure can be used to produce several eigenvalues and each eigenvalue creates one equation.

For a quasi-Newton approach to solving a nonlinear system [38], the most difficult part is that the Jacobian matrix is hard

Figure 1b A sample IVUS image of inner artery with atherosclerotic plaque. Courtesy of Wikipedia.

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to obtain accurately, which easily causes the divergence of the algorithm, since the cut-off errors in the matrix quickly accumulate and perturb the algorithm seriously.

Another scheme called optimization [39,40] is utilized to avoid the disadvantage. Define a function for optimization

( ) = | ( , ) |,i

S f D fπ λ∑ (4.1)

where ∑ denotes the summation sign. If f is the solution, it is also a local minimum of ( )S f and then ( ) = 0S f . A significant advantage of this approach is that one can over-estimate this problem (the quasi-Newton scheme is incapable of realizing this), which means the number of equations is bigger than the number of unknowns. The extra information helps find the solution more accurately and the algorithm becomes more robust.

There are several methods for finding local minima of a function, one of which is the Nelder-Mead simplex method [39,40]. It minimizes a function by using only function values without computing derivatives. The syntax in Matlab called fminsearch uses the Nelder-Mead simplex method as a line search for minimization and can be directly employed in the algorithm.

PERSPECTIVE AND CONCLUSIONSturm-Liouville problems arise almost throughout applied

mathematics. Inverse SLPs have received increasing attention because of the applications in physical sciences and particularly their emerging applications in biomedical engineering.

Inverse SLPs display a wide variety of formulations. The input information can come from either eigenvalues or eigenfunctions. Enough information is required for uniquely determining the unknown function incorporated inside the coefficients of the SLPs. Many classical inverse SLPs have been thoroughly studied analytically to show the existence and uniqueness, or more specifically, under what conditions one can prove the existence and uniqueness. However, the existent inverse SLPs are only based on simple formulations. Other more complex inverse SLPs remain untouched. Global existence and uniqueness need further investigation for these critical problems.

Some open questions arise. For example, if the input is from partial information of the eigenfunctions, are there other possibilities to extract the information? How can one prove the existence and uniqueness based on the new information? If half of the unknown function is given and the input information can only be obtained from the eigenfunctions, can we still get existence and uniqueness? More generally, what are the weights of eigenvalues and eigenfunctions in the inverse SLP respectively? Except the dual operator inverse SLPs, how do we deal with inverse SLPs with three, four or even higher number of operators?

Numerical approaches for recovery of the unknown functions were designed based on these theoretical investigations or newly invented numerical techniques. To fully recover the unknown, complete input information from an infinite set of eigenvalues or eigenfunctions is required. The information corresponds to the modes of the fourier expansion of the unknown function. It is natural for us to only recover the fourier modes of the unknown function. Practically, one can at most obtain finite counts of eigenvalues or eigenfunctions, from which one can recover an

enough-to-use approximation of the unknown. The unknown can also be approximated by spline techniques, under which we only need to recover the nodal values. It deserves attention that the numerical procedures for inverse SLPs are flexible. Some inverse SLP numerical techniques may not suit some special purposes. We then need to create new techniques for these special problems.

Inverse SLPs can be applied broadly in physical sciences, such as the vibrating strings, energy levels for quantum mechanics, and so on. It is also promising for application in more engineering areas because of its advantage of understanding or predicting the physical properties from easy-to-handle approaches. Broadly, inverse SLPs can be applied to any problem related to eigenfrequencies. The SLP form can be in a simple one (1.5) or in a more complicated form (1.1). For the later case, the unknown may not be the coefficients themselves. Instead, it may be involved in the coefficients through a more complicated way. The theoretical analysis thus becomes more difficult to be fully resolved. Theoretical proofs for the existence and uniqueness are hard to obtain. However, one can use numerical approaches to show them indirectly [8]. More specifically, recovery of the discretized unknown function equals solving a series of nonlinear equations. Thus, the existence can be numerically verified by observing the satisfaction of the equalities. The local uniqueness can also be checked by calculating the Jacobian matrix of these equations. If the determinant of the Jacobian matrix is not zero, it then implies uniqueness of the solution.

ACKNOWLEDGEMENTSZi Chen acknowledges the support from the Society in Science-

Branco Weiss fellowship, administered by ETH Zürich. Special thanks are given to Eric Dai and Jayne I. Hanlin for reading and comments about the article.

A. STURM-LIOUVILLE CONVERSION The second-order differential equation (1.4) can be converted

into the form (1.1) by virtue of the relations

( ) = ( ( ( ) / ( )) ),expx

p x H F dχ

ω ω ω∫ (A.1a)

( )( ) = ( ( ( ) / ( )) ),( )

expxG xq x H F d

F x χω ω ω− ∫ (A.1b)

( ( ( ) / ( )) )( ) = ,

( )

expx

H F dW x

F xχ

ω ω ω∫ (A.1c)

where [ , ]a bχ∈ is any constant.

B. MULTI-DIMENSIONAL SECANT METHODThe multidimensional secant method [41] attempts to find

the root of n linear or nonlinear equations with n unknowns

1 2( , ,...... ) = 0, =1,2 .i ng x x x for i n

Similar to the one dimensional secant method, one needs 1n+ trial solutions to start the method and denotes them by the

following identities ( ) ( ) ( )1 2= ( , ,...... ), =1,2 1.j j j

j nX x x x for j n+

Set 1n+ numbers 1Π , 2Π ,... 1n+Π to satisfy the following 1n+ equations

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1

=1

1( ) ( ) ( )1 2

=1

=1,

( , ,...... ) = 0 =1,2,... .

n

jj

nj j j

j i nj

g x x x for i n

+

+

Π

Π

∑

∑

Solve for 1Π , 2Π ,... 1n+Π to form a new vector 1* ( )=1

= n jjj

X X+Π∑ .

Then *X replaces the one among the 1n+ trial solutions which makes the summation of all the ig the greatest. As =1n , the method reduces to the one dimensional secant method.

The benefit of this approach is that enough initial information is provided for the algorithm such that a better convergence is guaranteed. The initial estimate of these trial solutions come from an educated guess.

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Gou K, Chen Z (2015) Inverse Sturm-Liouville Problems and their Biomedical Engineering Applications. JSM Math Stat 2(1): 1008.

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