University of Waterloo, Waterloo, Ontario University of ...

Optimized Cranial Bandeau Remodeling ∗

James Drake1, Marina Drygala2, Ricardo Fukasawa2, Jochen Koenemann2, AndreLinhares4, Thomas Looi1, John Phillips1, David Qian2, Nikoo Saber3, Justin Toth1,*, Chris

Woodbeck1, and Jessie Yeung1

1The Hospital for Sick Children, Toronto, Ontario2University of Waterloo, Waterloo, Ontario

3MyJiko, Toronto, Ontario4University of Waterloo, Waterloo, Ontario. Now at Google Research

*Corresponding Author: Justin Toth, [email protected]

March 16, 2021

Abstract

Craniosynostosis, a condition affecting 1 in 2000 infants, is caused by premature fusing of cranial vaultsutures, and manifests itself in abnormal skull growth patterns. Left untreated, the condition may lead tosevere developmental impairment. Standard practice is to apply corrective cranial bandeau remodelingsurgery in the first year of the infant’s life. The most frequent type of surgery involves the removal of theso-called fronto-orbital bar from the patient’s forehead and the cutting of well-placed incisions to reshapethe skull in order to obtain the desired result.

In this paper, we propose a precise optimization model for the above cranial bandeau remodelingproblem and its variants. We have developed efficient algorithms that solve best incision placement, andshow hardness for more general cases in the class. To the best of our knowledge this paper is the first tointroduce optimization models for craniofacial surgery applications.

Keywords: OR in medicine, Dynamic programming, Combinatorial optimization, Surgical planning

∗The authors were supported by the NSERC Discovery Grant program and acknowledge the support of the Natural Sciencesand Engineering Research Council of Canada (NSERC).

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Figure 1.1: (a) the skull of a craniosynostosis patient [18], (b) the schematics of a bandeau [16], and (c) abandeau attached to metal template [10]

1 Introduction

While adult skulls are a single solid piece of bone, infant skulls are in fact composed of several bone piecesthat start to fuse at the joints (cranial vault sutures) as the infant grows. However, when these sutures fuseprematurely, it leads to a condition called “craniosynostosis”.

The purpose of this work is to study optimization models arising from the treatment of craniosynostosis,which affects 1 in 2000 infants [17]. Craniosynostosis often yields abnormal growth patterns of the infant’sskull, ultimately resulting in severe deformations (see Figure 1.1.(a)). Left untreated, it may also lead toincreased intracranial pressure, and this in turn may cause stunted mental growth, vision impairment andseveral other severe impairments of the patient. That said, the primary reason for treatment is psychoso-cial concerns later in life from the abnormal skull shape. To avoid such problems, surgical treatment ofcraniosynostosis is often applied within the first year of an infant’s life [14].

To better understand the optimization problem that arises in such surgical treatment, we first focus onone of the methods to treat the condition. A central component in such surgeries is the reshaping of the so-called fronto-orbital bar, or bandeau, which is a strip of bone from the patient’s forehead above the eyebrows(see Figure 1.1.(b)). The surgeon removes the bandeau from the patient’s skull, then performs several well-placed incisions (‘kerfs’ [1]) into this piece of bone in order to make it more malleable. After kerfing thebone and the bandeau is fit perfectly into the template fixation is achieved using bioresorbable plates andscrews.( see Figure 1.1(c)). The reshaped bandeau is then inserted back into the patient’s forehead. Notethat in the bandeau procedure, the pieces relative position to each other is preserved. A second reshapingprocedure is one over the full vault, where larger pieces of the whole cranium are cut and may be rearrangedand put into completely different positions, so that the original arrangement and orientation of the piecesneed not be preserved.

The purpose of these reshaping procedures is to make the bone look closer to what is considered an idealshape. Thus, one important part of such problems is the determination of the ideal skull shape. Recently,a team of surgeons and researchers at the Hospital for Sick Children developed a library of normativepediatric skulls modelling ideal skull shapes of infants of varying ages [16]. Based on these skulls, thehospital now has such steel skull templates and the surgeon chooses the template that is most appropriateto each individual craniosynostosis patient, based on the patient’s age and size. The availability of suchindividual steel templates during operation has led to a more standardized, objective and precise correctionof craniosynostosis [16].

A key factor in adequately approximating the ideal skull shape is to strategically place the kerf points onthe deformed bandeau. This important part of the surgery is currently performed based on ad hoc methods,using the surgeon’s experience, intuition and talent. Obviously, this means that the quality of the outcomeof this procedure depends integrally on the surgeon’s experience and skill. Another major drawback is thecurrent difficulty in training new surgeons, and ensuring standardized and consistent surgical outcomes.

In this paper we focus on the surgeon’s task of finding locations for the fronto-orbital bar incisions, the

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bandeau reshaping problem (BRP). We define a formal mathematical problem, the curve refitting problem(CR), which models this task and allows us to evaluate the quality of an operation’s outcome. We present anefficient algorithm for this problem, and show hardness results for a variant of it. In particular, the variantstudied is motivated by the full-vault surgery and implies a hardness result in the full-vault problem. To thebest of our knowledge, our work presents the first formal injection of optimization methods into the realmof plastic craniofacial surgery.

1.1 Related Work

The use of optimization in healthcare has received increased attention in recent years. The range of ap-plications has gone from staff and facilities scheduling (e.g. [2, 3]) and outpatient scheduling [4] to cancerradiation treatment [11, 7]. However, to the best of our knowledge, this is the first work that applies anoptimization approach to a problem that deals with bone reconstruction. Indeed, most of the quantitativework in craniosynostosis treatment has focused on measuring the severity of cases [15, 6].

There have also been some similar problems studied in different contexts. For instance, [12] looks at theproblem of reconstructing an artifact based on its fragments. However, such work is based on reconstructinga previously known artifact based on how similar the edges of the broken pieces are, which does not apply inour case. In addition, there are works on computational geometry that study how to best match two sets ofpoints, like the Iterative Closest Point problem [5] and the Procrustes problem [9]. However these problemsonly allow some very well-defined uniform transformations (e.g. scaling, rotating) to be applied to the set ofpoints and thus cannot be applied in our setting, where one seeks to maintain some parts of the skull intact,while transforming (rotating) other parts, which are obtained after the incisions are done, independentlyfrom the remainder of the skull.

Therefore, none of the previous literature found would apply in solving the desired (BRP) and thus, thereis a need to develop new methodology to solve this problem.

1.2 Our Contributions and Outline

We consider the main contribution of our work to be the introduction of formal discrete optimization meth-ods for craniofacial plastic surgery. Moreover, to the best of our knowledge we are the first to provide aformal abstract model (CR) of the craniofacial surgical problem of reshaping the fronto-orbital bar, whichis presented in Section 2. We then present an exact algorithm for a practically important special case of theproblem based on dynamic programming in Section 3, along with the presentation of an efficient implemen-tation of the algorithm and empirical evaluation in Section 4. Lastly, we prove in Section 5 that the generalform of the problem is computationally hard.

2 Computational Models

In this section we present a mathematical model of the Bandeau Reshaping Problem (BRP). The modelis very general, accommodating many different cost functions for measuring dissimilarity between the idealand deformed curves. It also models surgeries in which the pieces of the deformed curve can be rearranged.This is not done in most (BRP) cases in practice, but is studied as a first step in trying to solve the moregeneral full-vault problem. In Subsection 2.2 we discuss a specialization of our model which does not admitrearrangement, and is thus more indicative of the bandeau surgery.

2.1 Curve Reshaping Problem

In a Curve Reshaping Problem (CR) we are given, as inputs, two piecewise linear continuous functions1

(henceforth referred to as curves) of the form f : [0, s] → R and g : [0, t] → R. The curves are used torepresent bandeaus from a top-down perspective, with curve f for the deformed bandeau and curve g for theideal bandeau. When presenting a curve η : [0, u] → R, we give a set of points D ⊆ [0, u] to the algorithm

1The requirement that our curves are functions is not necessary for our algorithm but greatly simplifies notation.

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as input, such that η is a piecewise linear interpolation of the list [(x, η(x)) : x ∈ D]. The set of points D iscalled the discretization of η.

We are also given as input to (CR) an integer k ≥ 1, indicating the number of cuts on f we intend tomake. The last piece of input we are given is a function c : 2[0,s] × [0, t] × [0, t] → R+ ∪ {∞}. For anya < b ∈ [0, s], l < r ∈ [0, t], the value of c([a, b], `, r) measures the cost of using the segment f

∣∣[a,b]

of f , in

order to cover a segment of g by clamping the left endpoint of f∣∣[a,b]

at ` and the right endpoint at r. See

Subsection 4.1 for an elaboration, where we discuss a particular cost function used in our target application.In our models we treat the function c as a black box oracle.

In a (CR) problem, the decisions are as follows: Choose k positions on the deformed curve, p1 < · · · <pk ∈ P, where P ⊆ [0, s]∩Q is the discretization of f . For simplicity we assume that p1 > 0 and pk < s andlet p0 = 0, pk+1 = s. These cuts segment f into pieces f

∣∣[p0,p1]

, . . . , f∣∣[pk,pk+1]

, which will be mapped onto

the ideal curve g. We index the resulting pieces of the deformed curve by the set {0} ∪ [k]. Here [k] is acommon shorthand notation for {1, . . . , k}.

Let Q ⊆ [0, t] ∩ Q be the discretization of g. We map the segments of f onto g by making the followingdecisions: for each i ∈ {0} ∪ [k], we select two positions ì, ri ∈ Q indicating positions of the left and rightendpoints that the ith segment of f will take on the ideal curve.

The objective in our (CR) problem is to minimize the total dissimilarity induced by our solution. Weintroduce the notion of coverage. Suppose

Q = {q1, . . . , q|Q|}.

For j ∈ [|Q| − 1] we say that the interval or segment [qj , qj+1] on g is covered by our solution if there is somei ∈ J such that

min(ì, ri) ≤ qj ≤ qj+1 ≤ max(ì, ri).

In this case piece i covers [qj , qj+1]. Let uncov(`, r) be the set of segments [qj , qj+1] on g that are notcovered by our solution. Note that since our clamp points ì, ri for piece i are elements of Q our notion ofcoverage includes any intervals of the ideal curve located between where we clamped the endpoints of ourcurve segment.

To penalize not covering a segment of the ideal curve we select a parameter δ ≥ 0 and add the term

δ|uncov(`, r)|

In summary, we may formally state our problem as follows:

min∑i∈J

c([pi, pi+1], ì, ri) + δ|uncov(`, r)| (CR)

s.t. p1 < · · · < pk ∈ P`, r ∈ Q{0}∪[k].

Notice that (CR) requires exactly k cuts to be made on the deformed curve. If one wanted find theoptimal solution with at most k cuts they can simply solve (CR) for different cut numbers from 0 to k.

2.2 Without Rearrangement

In a general 3-dimensional cranial vault remodeling surgery, where the entire skull is being reorganized,it is possible to take pieces and rearrange their relative positions. The (CR) model as described previouslyalso admits this possibility. In practice, the (BRP) surgery for reshaping the bandeau does not admit this.That is to say, the relative positions of the cut segments of the deformed curve are not changed by theirmapping onto the ideal curve. In fact the surgeons do not break apart the bandeau along the points wherethey cut it, rather they bend the bandeau into its proper shape at these points. To model this we define the(CR) problem without rearrangement. In this problem we add the constraints:

• ` ≤ r to indicate that the right endpoint of segment i needs to be placed further right than the leftendpoint of segment i, i.e. that segments cannot be flipped.

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• for all i ∈ [k], ri−1 = `i to indicate that the i-th segment is mapped to the right of the i−1-th segment.

In Section 3 we will give a polynomial time algorithm based on dynamic programming for the (CR)problem without rearrangement. This will be contrasted with a proof in Section 5 that the (CR) is stronglyNP-hard in general.

3 Dynamic Programming Algorithm

In this section, we present a dynamic programming algorithm for (CR) problems without rearrangement.For the remainder of this section, we consider an instance I of the (CR) problem without rearrangement,where the deformed curve f : [0, s]→ R and the ideal curve g : [0, t]→ R are discretized as point sets P andQ respectively. Let c be the associated cost function, δ ≥ 0 the uncoverage parameter, and let k ≥ 1 be aninteger indicating the associated number of cuts.

We intend to break instance I of (CR) into a set of suitable subproblems so that we can impose a dynamicprogramming paradigm. Choose an integer number of cuts k′ ∈ [0, k]. Choose a, b ∈ [0, s] ∩ P such thata ≤ b and |[a, b] ∩ P | ≥ k′ + 2. This is a subinterval on the deformed curve on which we can make k′ cutsexcluding the endpoints. Similarly choose d, e ∈ [0, t] ∩Q such that d ≤ e and |[d, e] ∩Q| ≥ k′ + 2. Now letCR([a, b]], [d, e], k′) denote the optimal value of (CR) problem without rearrangement using cost function c,on the deformed curve f

∣∣[a,b]

with feasible cut positions P ∩ [a, b] and ideal curve g∣∣[d,e]

with feasible clamp

positions Q ∩ [d, e] and following added constraints:

1. we restrict to feasible solutions which use exactly k′ cuts, and

2. we restrict to feasible solutions for which uncov(`, r) is empty.

By our choice of parameters, which we call a feasible choice of parameters, CR([a, b], [d, e], k′) always has afeasible solution. Our goal is to give a recursive formula for CR([a, b], [d, e], k′) that can be used to design adynamic program.

Theorem 3.1 For any feasible choice of parameters the following recursive equation holds

CR([a, b], [d, e], k′) =

=

min(p,q)∈P∩(a,b)×Q∩(d,e)

(c([a, p], d, q) + CR([p, b], [q, e], k′ − 1)) , if k′ > 0

c([a, b], d, e) otherwise.

Proof: If we expand the recursive expression on the right hand side of the equation, we obtain the costof a feasible solution to the problem for which CR([a, b], [d, e]), k′) is the optimal value, and hence the righthand side expression is an upper bound on the optimal value CR([a, b], [d, e], k′). It remains to show it isalso a lower bound.

We proceed by induction on k. If k′ = 0 then there is no decision to make. The only feasible solution isto map f

∣∣[a,b]

onto g∣∣[d,e]

and pay cost c([a, b], d, e) as desired.

Hence we may assume k′ > 0 and the inductive hypothesis holds on all CR(·, ·, k′′) where k′′ < k′. Letp∗1, . . . , p

∗k′ ∈ P ∩ (a, b) and (`∗1, r

∗1), . . . , (`∗k′ , r

∗k′) ∈ Q2 ∩ (d, e)2 be an optimal solution to CR([a, b], [d, e], k′).

Observe that

CR([a, b], [d, e], k′) =

k′∑i=1

c([p∗i−1, p∗i ], `∗i , r

∗i ) = c([a, p∗1], d, r∗1) +

k′∑i=2

c([p∗i−1, p∗i ], `∗i , r

∗i )︸︷︷︸

≥CR([p∗1 ,b],[r∗1 ,e],k

′−1)

≥ min(p,q)∈P∩(a,b)×Q∩(d,e)

(c([a, p], d, q) + CR([p, b], [q, e], k′ − 1)) .

The underbrace inequality above follows since p∗2, . . . , p∗k and (`∗2, r

∗2), . . . , (`∗k′ , r

∗k′) is a feasible solution to

the (CR) problem without rearrangement using exactly k′ − 1 cuts on curves f∣∣[p∗1 ,b]

and g∣∣[`∗2 ,e]

which pays

no uncoverage penalty. Note that since this problem is without rearrangement, `∗1 < r∗1 and r∗1 = `∗2.

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By Theorem 3.1, when δ = 0, I can be solved using standard dynamic programming techniques utilizing atable of size O(|P |2|Q|2k). Each table entry can be computed in O(|P ||Q|) time, yielding an algorithm solving(CR) without rearrangement in time O(|P |3|Q|3k). Since k is at most min{|P |, |Q|} this is a polynomialtime algorithm. In fact, the experiments in Section 4 demonstrate that a very small number of cuts areneeded relative to the size of P and Q in order to obtain an extremely high quality bandeau reconstruction.

For δ > 0, observe that the uncoverage penalty γ(p0 + (s − pk)) is completely determined once `0 andrk are fixed because we are working in the paradigm which disallowed rearrangement. Since ri−1 = li, andl <= r, the only possible uncovered parts of g are the ones on points less than l0 or greater than rk, whichgives precisely the uncoverage penalty above. Then it remains to solve a (CR) problem with optimal valueCR([0, s], [`0, rk], k). By enumerating our choices for `0, rk and solving the subproblems using Theorem 3.1,we get the following corollary:

Corollary 3.2 Any (CR) problem without rearrangement can be solved in time polynomial in |P |, |Q| andlinear in k.

4 Computational Experiments

In this section, we describe results obtained by testing our dynamic programming algorithm for (CR) withoutrearrangement on a test suite of craniosynostosis cases. Due to medical privacy regulations and institutionalpolicy it is difficult and time-consuming to obtain scans of actual patient skulls from operations performedat the Hospital for Sick Children. We were able to obtain three such test cases including both pre-op andpost-op scans with all patients’ personal identifying information removed. Results from the tests performedwith those are described in Subsection 4.2. To augment these results we designed a mathematical modelto generate a large number of synthetic test cases. In Subsection 4.3 we describe the model and test casesresulting from it. In Subsection 4.4 we report on the results of applying our algorithm to the test casesgenerated by our synthetic model. Subsection 4.1 discusses the particular choice of cost function for problem(CR) we implemented in our tests.

For our tests we ran an implementation of algorithm discussed in Section 3 written in Python 3.6.7 usingNumpy 1.15. The tests were run on a Macbook Pro with a 2.6GHz 6-Core Intel i7 processor and 16GB ofDDR4 RAM, running macOS Catalina version 10.15.7.

4.1 Computing the Dissimilarity

Lf

Rf

(a) Curve f

Lg Rg

(b) Curve g

(c) d(f, g)

Figure 4.1: Calculating d(f, g)

To evaluate the quality of a solution, we define, forany pair of curves f : [0, s]→ R and g : [0, t]→ R, adissimilarity measure d(f, g) ∈ R ∪ {∞}. Note thatthe dissimilarity measure is not necessarily symmet-ric, that is, the measures d(f, g) and d(g, f) are notnecessarily equal. Before proceeding we reiteratethat the algorithm does not rely on any propertiesof d, and hence can be applied to different measuresat the discretion of the user.

We will use Lf and Rf as shorthands for theleft and right endpoints (0, f(0)), and (s, f(s)) of f ,respectively. We say that two curves f and g matchif the Euclidean distance between the endpoints off is approximately equal to the Euclidean distancebetween the endpoints of g. Formally, we introduce

a parameter α ∈ [0, 1], and we say that f and g match if 1− α ≤ ‖Lg−Rg‖2‖Lf−Rf‖2

≤ 1 + α .

If f and g do not match, then we define d(f, g) = ∞. Next, we describe how to compute d(f, g)when they do match. First, we rotate g around Lg so that its right endpoint is moved to the point(x(Lg) + ‖Lg −Rg‖2 , y(Lg)

), and we denote by g the curve thus obtained. Next, we modify the curve

f as follows. We scale it by a factor of‖Lg−Rg‖2‖Lf−Rf‖2

, so that the Euclidean distance between its endpoint

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2153895 Pre-Op and Ideal

-15

0

15

30

45

60

-60 -45 -30 -15 0 15 30 45 60

IdealPre-Op

2153895 Post-Op and Ideal

-17.5

0

17.5

35

52.5

70

-70 -52.5 -35 -17.5 0 17.5 35 52.5 70

IdealPost-Op

Figure 4.2: A top-down plot comparing Pre-Op and Post-Op Curves to Ideal

becomes equal to the Euclidean distance between the endpoints of g. Next, we translate and rotate it so thatits left and right endpoints are mapped to Lg and Rg respectively. We denote by f the curve thus obtained.

Finally, we define d(f, g) to be the area between the modified curves f and g. If f and g are functions we

can write d(f, g) =∫ x(Rf )

x(Lf )|f(x)− g(x)| dx .

Unfortunately we cannot guarantee that f and g are functions, so we draw on methods in computationalgeometry. Since f and g are piecewise linear and their endpoints line up, their union forms the boundary ofa closed polygon A. By computing where the edges of A intersect we can partition A into a set of simplepolygons in time bounded by a polynomial in the number of points in the discretization. Using the well-known Shoelace Method [13] we can compute the area of each such simple polygon and sum them to obtainthe area of A. This area we denote d(f, g). Figure 4.1 shows an example of two curves f and g and howd(f, g) is calculated.

Now we are ready to define a cost function c : 2[0,s] × [0, t]× [0, t]→ R ∪ {∞} for the (CR) problem. Forall [a, b] ⊆ [0, s] and all [`, r] ⊆ [0, t], we define

c([a, b], `, r) = d(f |[a,b], g|[`,r]).

Note that curves presented via an explicit discretization, such as those presented to the DP algorithmimplied by Theorem 3.1 are piecewise linear functions whose inflection points are all part of the input. Thusevaluating d on pairs of such curves can be done in polynomial time (in the number of pieces), and hencec can be queried in polynomial time. In our runs of the algorithm, we pre-compute a table of costs c andassume O(1) access to its elements during operation of the dynamic program.

4.2 Comparing Our Algorithm to Surgical Outcomes

The Hospital for Sick Children provided us with 3d images of three pre-operation (pre-op) and three corre-sponding post cranial vault remodelling operation (post-op) skulls of metopic2 cases where they performedsurgical interventions. The post-op scans are taken between 3-5 days after the surgery. From these scans wecreated deformed curves plotted from a top-down perspective as described in Section 2. Figure 4.2 showsthe resulting curves from one such case.

The curves are given to the algorithm as an explicit list of points describing a piecewise linear curve. Thediscretization was chosen so that the distance between two consecutive points is between 0.5 and 1mm. Thisresulted instances where the deformed curves had 200 coordinates. On these three instances our algorithmtook on average 0.323 seconds to find the optimal solution for up to 13 cuts. Figure 4.3 shows a plot of theoptimal value of the objective function (Area Below Curve or ABC, see Section 4.1) as a function of thenumber of cuts our algorithm applies for the cases provided by Sick Kids. The Post-Op ABC’s are providedto show when our algorithm crosses the ABC achieved by the surgeons. In each case, only three cuts areneeded for the algorithm to achieve a lower ABC than that achieved by the surgeon. Although the surgeonsdid not record the specific number of cuts made in an actual operation, looking at the data provided, thesurgeons estimate that at least 7 cuts were made to achieve the Post-Op curves.

In Figure 4.4 we see our algorithm’s surgical plan using three cuts for a case provided by Sick Kids. Thegreen curve is a top-down view of the ideal curve, and the red is a top-down view of the deformed curve. Theblue curve is the curve which results from performing the surgery as recommended by the algorithm. The

2Metopic is a medical term referring to a common variant of craniosynostosis.

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Figure 4.3: Cuts vs Area Below Curve (ABC) for Hospital For Sick Children Cases

red squares denote the cut positions on the deformed curve, and the blue circles denote the correspondingpositions on the ideal curve these cuts are mapped to.

4.3 Synthetic Craniosynostosis Model

In the interest of collecting a larger data set on which to test our algorithm we created a synthetic mathe-matical model of craniosynostosis cases which can be used to generate instances of (CR). Our model is basedon two observations. The first is that craniosynostosis cases are often characterized by a few distinct regionswhere the curve exhibits different properties. For example, in metopic craniosynostosis the premature fusionof the suture in the middle of the forehead leads to steep curve segments coming to a point near the origin,and more shallow curve segments as the points get further from the origin. The second observation is thatsurgeons at the Hospital for Sick Children choose the scale of the ideal bandeau so that the furthest pointsfrom the origin, as well as the origin, intersect the deformed curve at the furthest points from the origin andthe origin respectively. In most practical cases this is always possible.

Formally, for deformed point set P , the ideal point set Q is chosen from a collection of scaled ideal curvesso that

{arg min{x : (x, y) ∈ P}, (0, 0), arg max{x : (x, y) ∈ P}} ⊆ Q.

If we fix the scaling of the ideal curve then this means we have a specific set of identified coordinates weneed deformed curves to intersect, and in the regions between the intersecting points, the specific type ofcraniosynostosis case will dictate the steepest or degree of the polynomial representing the curve.

The ideal bandeau provided by Sick Kids is a smooth 2-dimensional curve intersecting the origin, withendpoints (−x, y) and (x, y) for some x, y ∈ R>0. The choice of coordinates (x, y) may vary as the bandeaucan be scaled to best fit each individual patient. The baseline bandeau without scaling we were given hadx = y = 50.0.

The Synthetic Craniosynostosis Model is defined as follows. Let (x0, y0) denote the positive endpoint ofideal curve g intersecting the origin. A synthetic deformed curve f can be defined by specifying two things.The first is a list of points in R2, L = {`1, . . . , `|L|}, which is monotonically increasing in the first coordinate

8

2153895 With 3 Cuts

-15

0

15

30

45

60

-60 -45 -30 -15 0 15 30 45 60

Ideal

Pre-Op

Post-Op

Pre-Op cut loca7ons

Post-Op cut loca7ons

Figure 4.4: Algorithm’s Surgical Recommendation for Case 2153895 With 3 Cuts

metopic6

-12.5

0

12.5

25

37.5

50

-50 -37.5 -25 -12.5 0 12.5 25 37.5 50

IdealPre-Op

Figure 4.5: A Synthetic Metopic Case with a split-points at (0, 0) and (±12.5, 7).

and contains (−x0, y0), (0, 0), and (x0, y0). The second is a list of rational numbers, D = {d1, . . . , d|L|−1},of size |L| − 1. For any x ∈ [−x0, x0], the value of f(x) is defined to be

f(x) := aixdi + bi, where i ∈ {1, . . . , |L| − 1} such that x ∈ [ì0, `

i+10 ]

where each ai and bi are chosen to satisfy the equations

ai(ì0)di + bi = ì1

ai(ì+10 )di + bi = ì+1

1 .

Intuitively, f is a piecewise curve consisting of |L| segments, where the i-th segment is chosen as a curveof the form axdi +b such that it intersects the i-th and i+1-th points in the list L. The example in Figure 4.5shows a synthetic deformed curve with

L = {(−49.3, 48.7), (−12.5, 7), (0, 0), (12.5, 7), (49.3, 48.7)}

and D = {2, 1, 1, 2}.The attentive reader will notice that f defined this way is not piecewise linear. To make the curve given as

input to a (CR) problem piecewise linear, we discretize it by sampling a set of points D ⊆ [`1, `|L|]×range(f)uniformly spaced out in their first coordinate and approximating f by the piecewise lienar interpolation of D.Of course, the greater the size of D the better the approximation of f . In our examples we chose |D| = 200since the ideal curves we were working with also had a discretization using 200 points.

9

sagi%al23

-12.5

0

12.5

25

37.5

50

-50 -37.5 -25 -12.5 0 12.5 25 37.5 50

IdealPre-Op

Figure 4.6: A Synthetic Sagittal Case

4.4 Synthetic Model Test Results

Using the Synthetic Craniosynostosis Model discussed in Subsection 4.3 we created three buckets of test casesto reflect different types of craniosynostosis. The data can be inspected at https://github.com/wjtoth/optimized-cranial-bandeau-remodeling.

The first bucket consisted of 24 synthetic metopic cases. To create these cases, for each test case wesampled an (x, y) coordinate in [12.5, 25]× [2, 20] and chose L to be

{(−49.3, 48.7), (−x, y), (0, 0), (x, y), (49.3, 48.7)}

and we sampled two integers d1 ∈ {1, 2} and d2 ∈ {2, 3} and chose D to be {d2, d1, d1, d2}. This createscurves where the split-point (x, y) lies above the ideal curve, and the segments intersecting the origin aresteeper than the segments which do not. This simulates the premature fusing of suture in the middle of theforehead which characterizes metopic craniosynostosis. An example can be seen in Figure 4.5.

The second bucket consisted of 24 synthetic sagittal cases. These cases are characterized by a prematurefusing of the sagittal sutures [19], resulting in a skull shape where the forehead appears more flat and widefrom a top down perspective. To simulate this phenomenon we created each test case by sampling an (x, y)coordinate in [27, 55] × [0.5, 2] and two integers d1 ∈ {1, 2} and d2 ∈ {2, 4}. We constructed L and D fromthese samples in the same manner as we did for the metopic cases above. The result is test cases where thesplit-point lies below the ideal curve, and the curve is mostly flat until it reaches the split-point, then bendsup quickly to intersect the ideal endpoints. An example can be seen in Figure 4.6.

The last bucket consisted of 20 extreme cases. The purpose of these cases was to test the robustness of ouralgorithm. We emphasize that these cases are not reflective of what real life cases may look like. To generateeach case we sampled 5 random split-points (x1, y1) . . . (x5, y5) ∈ [1, 49]2 sorted in order of increasing x value,and sampled 6 random degrees d1, . . . , d6 ∈ {1, 2, 3, 4}. Then we chose L to be

{(−50, 50), (−x5, y5), . . . , (−x1, y1), (0, 0), (x1, y1), . . . , (x5, y5), (50, 50}

and chose D to be {d6, . . . , d1, d1, . . . , d6}. An example can be seen in Figure 4.7.After discretization all of our synthetic test cases had on the order of 200 points. The average computation

time for the dynamic program in the metopic bucket was 0.13 seconds, in the sagittal bucket it was 0.11seconds, and in the extreme bucket it was 1.12 seconds. We note that the computational times are verylow, even in the extreme cases, and thus we consider that for all practical purposes, the running times areacceptable from the application point of view. The 25th, 50th, and 75th percentiles for ABC as functionof Number of Cuts used by the algorithm for the metopic, sagittal, and extreme buckets are presented inFigure 4.8. In each bucket we can observe a sharp decline in Area Below the Curve, and our algorithm needingonly 3 cuts to remove over 75% of the ABC. This rapid decay suggests that rearrangement is not neededin this application to get good outcomes, providing a principled justification for not using rearrangement insurgerical procedures. We conclude that the algorithm is fast, robust, and does better than current practicein terms of quality of outcome as measured by our objective function.

10

https://github.com/wjtoth/optimized-cranial-bandeau-remodeling

https://github.com/wjtoth/optimized-cranial-bandeau-remodeling

Figure 4.7: A Synthetic Extreme Case

Figure 4.8: Percentile curves for Synthetic Cases

11

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Ideal


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Deformed

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Peak<latexit sha1_base64="S7YmDDcyrahJRmD1wu8jfG8eezQ=">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</latexit><latexit sha1_base64="qBi/Ez+uvQoiOY3gB+T0WcHSNAI=">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</latexit><latexit sha1_base64="qBi/Ez+uvQoiOY3gB+T0WcHSNAI=">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</latexit><latexit sha1_base64="aJZxrnHlfL1fHEKKSVOimwpIdus=">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</latexit>

Bucket1<latexit sha1_base64="lBxsiryrAHEnAlsOjC1CldlLHwc=">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</latexit><latexit sha1_base64="LR7j/q4Qn4AYTW+3yHvWDM+rLpM=">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</latexit><latexit sha1_base64="LR7j/q4Qn4AYTW+3yHvWDM+rLpM=">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</latexit><latexit sha1_base64="W5pvJ7iBxelx2S0rfQ4EcmkWDZ0=">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</latexit>

Figure 5.1: The instance of (CR) generated from an instance of 3-Partition with m = 2, a1 = 1, a2 = a3 =a5 = 2, a4 = 4, a6 = 3 by our reduction.

5 Hardness in Model with Rearrangement

The (CR) problem with rearrangement involves cutting and rearranging pieces along the length of thebandeau. This problem is of interest as we wish to gain an understanding of the cranial vault remodelingprocedure, which involves cutting and rearranging segments of bone over a larger portion of the patient’sskull. We will show that the (CR) problem with rearrangement is strongly NP-hard. We will use a reductionfrom the strongly NP-complete 3-Partition problem [8].

In an instance of 3-Partition the input is a set A = {a1, . . . , a3m} of 3m elements for a positive integer m.In addition we are given a weight function s : A → Z≥0 such that B

4 < s(a) < B2 for all a ∈ A, such that∑

a∈A s(a) = mB where B is a positive integer. The goal is to decide if there exists a partition of A into msets A1, . . . , Am such that for each i ∈ {1, . . . ,m} we have that

∑a∈Ai

s(a) = B. Note that the choice of s,implies that if such a partition exists then |Ai| = 3 for each i ∈ [m].

To prove (CR) problem with rearrangement is strongly NP-hard by Lemma 5.1, we can give a pseudo-polynomial time transformation from 3-Partition.

Lemma 5.1 [8] If Π is strongly NP -complete and there exists a pseudo-polynomial time transformationfrom Π to Π′, then Π′ is strongly NP -hard.

The idea behind our proof is to take an instance of 3-Partition and create a deformed curve which consistsof a flat line whose discretization segments the line into 3m pieces such that the i-th piece has length s(ai),and create an ideal curve which consists of m “buckets”: length m flat line segments discretized uniformlyinto unit length segements. The buckets will be separated by tall “peaks” which make it impossible for adeformed segment corresponding to a 3-Partition element to be placed in multiple buckets without incurringsome cost. See Figure 5.1 for an example of this reduction.

Theorem 5.2 The (CR) with rearrangement is strongly NP-hard, even if we restrict to cost functions forwhich mapping a deformed segment onto an ideal segment incurs 0 cost if and only if the deformed and idealsegments are equal up to translation and rotation of their endpoints.

Proof: Consider an instance I1 of (A, s) of 3-Partition. Let |A| = 3m and let B = 1ms(A). We may assume

m ≥ 2 otherwise I1 is trivial. Note that from the definition of 3-Partition B is a positive integer.From I1 we will construct an instance I2 of (CR). For i = 1, . . . , 3m let Si =

∑ij=1 s(aj) be the prefix

sum of the sizes of the first i elements of A. Let S0 = 0. The deformed curve in our (CR) instance will be apiecewise linear interpolation of the discretization P ⊆ R2 determined by 3 sets P1, P2, P3 as follows:

P = P1∪P2∪P3

12

whereP1 = {(Si, 0) : i = 0, 1, . . . .3m},

P2 = {(s3m + 2i+ 1, 2B) : i = 0, . . . ,m− 2},

P3 = {(s3m + 2i, 0) : i = 1, . . . ,m− 1}.

Intuitively f is a piecewise linear function which is the constant 0 function from 0 to s(A) followed by m− 1“zigzags” between 2B and 0 where each peak and dip are one unit apart. We call these objects peaks ofheight 2B.

The constant 0 section of f is discretized into 3m segments by P where the i-th segment is of lengths(ai). We call that segment the segment corresponding to ai. The purpose of the peaks is to fit exactly ontothe barriers in the ideal curve which we will describe now.

The ideal curve g is the piecewise linear interpolation of discretization Q ⊆ R2 determined by 2 setsQ1, Q2 as follows:

Q = Q1∪Q2

where

Q1 = Z ∩m−1⋃i=0

[(B + 2)i, (B + 2)i+B],

Q2 = {((B + 2)i+B + 1, 2B) : i = 0, . . . ,m− 2}.

Intuitively Q1 is a discretization of m constant functions of value 0 of length B such that they are discretizedinto B unit length segments. We call each of these segments a bucket. Q2 is a set of m− 1 peaks of height2B which separate the m flat buckets which form Q1.

Choose any δ > 0 as the uncoverage parameter for I2. Choose any cost function c such that for segmentsany pair of segments (a, b) ⊆ P×Q the cost of assigning a to b, c(a, b), is 0 if and only if a and b are equal up totranslation and rotation of their endpoints. Choose a number of cuts k ∈ Z equal to (3m+1)+2(m−1)−2 =|P | − 2.

Observe that our construction can be done in pseudopolynomial time. In particular P and Q2 bothconsist of O(m) points and m = O(|A|). This is a polynomial number of points. The largest section is Q1

which consists of O(mB) points. Since mB = s(A) this is a pseudopolynomial numbers of points in the sizeof I1.

Now we claim that I1 is a Yes instance of 3-Partition if and only if the optimal value of I2 is 0.First suppose that I1 is a Yes instance. Then I1 has a solution A1, . . . , Am. Since every point in P is

cut by our choice of k, we need to assign every line segment of f to a position on g. For each i ∈ [m], orderthe elements of Ai arbitrarily. Consider the i-th bucket on Q, i.e. the intersection of Q1 with the interval[(B + 2)i, (B + 2)i + B]. We can arrange the segments of P corresponding to Ai along this interval paying0 cost since

∑a∈Ai

s(a) = B, they are all flat line segments, and Q1 has cut points at every integer pointin the interval. To complete our 0 cost solution to I2, arrange the m− 1 peaks formed by P2 ∪ P3 onto them− 1 peaks formed by cutting Q2. These peaks are of the same height and number so clearly they incur 0cost as they match exactly. Hence we have found a 0 cost solution to I2.

Now on the other hand, if the optimal value of I2 is 0 then no part of Q is left uncovered, and everysegment of P which is assigned to Q matches exactly. Since the peaks of Q have non-zero height, the optimalsolution to I2 will incur positive cost if one endpoint of a segment of P corresponding to an element a ∈ A isassigned to a point in one bucket and its other endpoint is assigned to a point in a different bucket. Further,by the height of a peak, if a segment corresponding to an element of A is assigned to cover a peak of Q thenit will not match exactly and will incur a positive cost. Thus, since every peak of Q is covered, every peakof P must be assigned to a peak of Q in an optimal solution. Since segments of P which corresponding toelements of A cannot cross buckets they are each assigned to a bucket of Q. By the lengths of the segmentsand buckets there can be no overlap, otherwise there will be some uncoverage (which would incur a positivecost). Hence the sum of the lengths of the segments of P1 assigned to the i-th bucket of Q is the lengthof the i-th bucket of Q, B. Construct a solution to I1 by choosing Ai to be the elements corresponding tosegments of P1 which are assigned to the i-th bucket of Q. Therefore I1 is a Yes-instance, as desired.

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Our proof of Theorem 5.2 actually shows that it is strongly NP-complete to decide if an instance of (CR)has a 0 cost solution. Thus we also obtain the following corollary about the hardness of approximation of(CR).

Corollary 5.3 There is no ν(|G|, |F |)-approximation algorithm for the (CR) problem unless P = NP , whereν is any function of |G|, |F | that is computable in polynomial time.

6 Conclusion

The work described in this paper is part of a larger push to improve craniofacial surgical methods throughapplied mathematics and engineering. The algorithm from Section 3 is integrated into a pre-operativeplanning tool, allowing surgeons to pre-plan cut locations. The output of the algorithm and the planningtool is designed to interface with a projection system that shows the cut locations directly on the patient inthe operating room. A prototype system is nearly complete at this stage.

In this paper we presented a formal optimization model, and an algorithm for the craniofacial surgicalproblem of reshaping the front-orbital bar. We demonstrated its application to several test cases. Our workon the 2D setting continues, and we are currently pursuing extensive clinical testing, and in particular apostoperative, and comparative evaluation of the quality of our solutions.

Future work will focus on 3D, anterior skull craniosynostosis cases, where surgical incisions and optimiza-tion problems are much more complex.

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https://en.wikipedia.org/wiki/Craniosynostosis

https://en.wikipedia.org/wiki/Sagittal_suture

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