OsteoFlow: Lyapunov-Guided Flow Distillation for Predicting Bone Remodeling after Mandibular Reconstruction

Hamidreza Aftabi; Faye Yu; Brooke Switzer; Zachary Fishman; Eitan Prisman; Antony Hodgson; Cari Whyne; Sidney Fels

cs.CVMar 23, 2026

OsteoFlow: Lyapunov-Guided Flow Distillation for Predicting Bone Remodeling after Mandibular Reconstruction

Authors: Hamidreza Aftabi, Faye Yu, Brooke Switzer, Zachary Fishman, Eitan Prisman, Antony Hodgson, Cari Whyne, Sidney Fels,

Organizations: Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, British Columbia, Canada · Sunnybrook Research Institute, University of Toronto, Toronto, Ontario, Canada · Department of Mechanical Engineering, University of British Columbia, Vancouver, British Columbia, Canada · Department of Surgery, University of British Columbia, Vancouver, British Columbia, Canada

Abstract

Predicting long-term bone remodeling after mandibular reconstruction would be of great clinical benefit, yet standard generative models struggle to maintain trajectory-level consistency and anatomical fidelity over long horizons, particularly in low-data regimes. We introduce OsteoFlow, a flow-based framework predicting Year-1 post-operative CT scans from Day-5 scans. Our core contribution is Lyapunov-guided trajectory distillation: Unlike one-step distillation, our method distills a continuous trajectory over transport time from a registration-derived stationary velocity field teacher. Combined with a resection-aware image loss, this enforces geometric correspondence without sacrificing generative capacity. Evaluated on 344 paired regions of interest, OsteoFlow significantly outperforms state-of-the-art baselines, reducing mean absolute error in the surgical resection zone by ~20%. This highlights the promise of trajectory distillation for long-term prediction in low-data clinical settings. Code is available on GitHub: https://github.com/hamidreza-aftabi/OsteoFlow.

Figures & tables

Figure 1: Illustration of the preprocessing steps and the teacher–student Lyapunov-guided distillation framework for guiding the student velocity field.

Figure 2: Model predictions for three representative cases (union, partial union, and nonunion), shown on the resection plane and two other central orthogonal slices. The purple circle shows the partial union near the resection site, while the pink ellipsoid depicts the nonunion.

Method	Mid slab				Full volume
	Dice (%) $\uparrow$	MS-SSIM (%) $\uparrow$	MAE All HU $\downarrow$	MAE Bone HU $\downarrow$	Dice (%) $\uparrow$	MS-SSIM (%) $\uparrow$	MAE All HU $\downarrow$	MAE Bone HU $\downarrow$
Day5 $\leftrightarrow$ Year1	85.60 $\scriptscriptstyle\pm 0.18$	64.18 $\scriptscriptstyle\pm 0.36$	95.89 $\scriptscriptstyle\pm 2.93$	177.94 $\scriptscriptstyle\pm 5.40$	94.36 $\scriptscriptstyle\pm 0.12$	87.13 $\scriptscriptstyle\pm 0.26$	41.78 $\scriptscriptstyle\pm 1.75$	70.32 $\scriptscriptstyle\pm 3.27$
ResUNet++ [ 25 ]	92.65 $\scriptscriptstyle\pm 0.11$	78.10 $\scriptscriptstyle\pm 0.22$	56.85 $\scriptscriptstyle\pm 1.64$	102.67 $\scriptscriptstyle\pm 4.20$	96.57 $\scriptscriptstyle\pm 0.08$	91.82 $\scriptscriptstyle\pm 0.17$	27.76 $\scriptscriptstyle\pm 1.12$	45.80 $\scriptscriptstyle\pm 2.50$
Pix2Pix [ 26 ]	92.51 $\scriptscriptstyle\pm 0.11$	78.26 $\scriptscriptstyle\pm 0.25$	56.96 $\scriptscriptstyle\pm 1.80$	103.55 $\scriptscriptstyle\pm 4.50$	96.45 $\scriptscriptstyle\pm 0.08$	91.86 $\scriptscriptstyle\pm 0.18$	27.75 $\scriptscriptstyle\pm 1.26$	46.38 $\scriptscriptstyle\pm 2.71$
GRIT [ 27 ]	92.45 $\scriptscriptstyle\pm 0.13$	77.05 $\scriptscriptstyle\pm 0.31$	60.07 $\scriptscriptstyle\pm 2.21$	100.63 $\scriptscriptstyle\pm 4.10$	96.48 $\scriptscriptstyle\pm 0.09$	91.32 $\scriptscriptstyle\pm 0.22$	30.31 $\scriptscriptstyle\pm 1.56$	45.78 $\scriptscriptstyle\pm 2.63$
cDDPM( $\Delta$ ) [ 28 , 29 ]	93.64 $\scriptscriptstyle\pm 0.15$	80.02 $\scriptscriptstyle\pm 0.34$	50.45 $\scriptscriptstyle\pm 2.44$	87.15 $\scriptscriptstyle\pm 4.81$	97.01 $\scriptscriptstyle\pm 0.11$	92.21 $\scriptscriptstyle\pm 0.24$	26.14 $\scriptscriptstyle\pm 1.62$	39.71 $\scriptscriptstyle\pm 3.19$

Table 1: Baseline comparison (mean

\pm

std across runs). Among non-teacher methods, the best and second-best results are shown in bold and underlined , respectively. An asterisk (*) indicates a significant difference from the second-best result using a two-sided Wilcoxon signed-rank test on paired ROI-level test metrics (

p<0.05

R

A

I

T_{1}

T_{2}

W

Mid slab

Full volume

Dice (%)

\uparrow

MS-SSIM (%)

\uparrow

MAE All HU

\downarrow

MAE Bone HU

\downarrow

Dice (%)

\uparrow

MS-SSIM (%)

\uparrow

MAE All HU

\downarrow

MAE Bone HU

\downarrow

Teacher Effect

✓

\times

\times

\times

\times

\times

88.97

\scriptscriptstyle\pm 0.20

66.76

\scriptscriptstyle\pm 0.48

82.47

\scriptscriptstyle\pm 3.08

139.84

\scriptscriptstyle\pm 5.86

95.25

\scriptscriptstyle\pm 0.13

87.45

\scriptscriptstyle\pm 0.30

40.07

\scriptscriptstyle\pm 1.92

63.89

\scriptscriptstyle\pm 3.96

\times

\times

\times

✓

\times

\times

90.79

\scriptscriptstyle\pm 0.17

71.23

\scriptscriptstyle\pm 0.41

77.00

\scriptscriptstyle\pm 2.74

123.85

\scriptscriptstyle\pm 5.33

95.72

\scriptscriptstyle\pm 0.11

89.05

\scriptscriptstyle\pm 0.27

37.08

\scriptscriptstyle\pm 1.70

55.91

\scriptscriptstyle\pm 3.58

✓

\times

\times

✓

\times

\times

91.54

\scriptscriptstyle\pm 0.16

73.20

\scriptscriptstyle\pm 0.39

69.79

\scriptscriptstyle\pm 2.41

115.66

\scriptscriptstyle\pm 5.02

96.00

\scriptscriptstyle\pm 0.10

89.68

\scriptscriptstyle\pm 0.26

34.51

\scriptscriptstyle\pm 1.49

52.91

\scriptscriptstyle\pm 3.37

✓

\times

\times

\times

✓

\times

89.67

\scriptscriptstyle\pm 0.22

69.22

\scriptscriptstyle\pm 0.50

79.29

\scriptscriptstyle\pm 2.96

137.79

\scriptscriptstyle\pm 5.71

95.49

\scriptscriptstyle\pm 0.14

88.48

\scriptscriptstyle\pm 0.33

37.60

\scriptscriptstyle\pm 1.83

60.03

\scriptscriptstyle\pm 3.80

Full Pipeline

✓

\times

\times

\times

\times

92.86

\scriptscriptstyle\pm 0.13

79.25

\scriptscriptstyle\pm 0.28

55.24

\scriptscriptstyle\pm 2.05

97.28

\scriptscriptstyle\pm 4.45

96.70

\scriptscriptstyle\pm 0.12

92.32

\scriptscriptstyle\pm 0.22

26.98

\scriptscriptstyle\pm 1.54

45.06

\scriptscriptstyle\pm 3.04

Table 2: Ablation study (mean

\pm

std across runs).

R

= rectified flow;

T_{1}

= our Lyapunov-guided distillation;

T_{2}

= MSE-based distillation;

A

= augmentation;

I

= image-space loss;

W

= resection-aware weight. Best results are in bold .

Backbone	$A$	$I$	Mid slab				Full volume
			Dice (%) $\uparrow$	MS-SSIM (%) $\uparrow$	MAE All HU $\downarrow$	MAE Bone HU $\downarrow$	Dice (%) $\uparrow$	MS-SSIM (%) $\uparrow$	MAE All HU $\downarrow$	MAE Bone HU $\downarrow$
SwinUNETR	✓	$\times$	92.30 $\scriptscriptstyle\pm 0.14$	77.79 $\scriptscriptstyle\pm 0.30$	58.83 $\scriptscriptstyle\pm 1.86$	103.92 $\scriptscriptstyle\pm 4.62$	96.58 $\scriptscriptstyle\pm 0.10$	91.44 $\scriptscriptstyle\pm 0.21$	28.28 $\scriptscriptstyle\pm 1.30$	46.82 $\scriptscriptstyle\pm 2.82$
Res-UNet	✓	$\times$	92.86 $\scriptscriptstyle\pm 0.13$	79.25 $\scriptscriptstyle\pm 0.28$	55.24 $\scriptscriptstyle\pm 2.05$	97.28 $\scriptscriptstyle\pm 4.45$	96.70 $\scriptscriptstyle\pm 0.12$	92.32 $\scriptscriptstyle\pm 0.22$	26.98 $\scriptscriptstyle\pm 1.54$	45.06 $\scriptscriptstyle\pm 3.04$
SwinUNETR	✓	✓	93.27 $\scriptscriptstyle\pm 0.13$	80.24 $\scriptscriptstyle\pm 0.27$	51.95 $\scriptscriptstyle\pm 1.78$	89.21 $\scriptscriptstyle\pm 4.36$	96.86 $\scriptscriptstyle\pm 0.10$	92.74 $\scriptscriptstyle\pm 0.20$	25.49 $\scriptscriptstyle\pm 1.22$	41.24 $\scriptscriptstyle\pm 2.94$
Res-UNet	✓	✓	93.65 $\scriptscriptstyle\pm 0.12$	81.39 $\scriptscriptstyle\pm 0.25$	49.47 $\scriptscriptstyle\pm 1.74$	83.65 $\scriptscriptstyle\pm 4.06$	97.00 $\scriptscriptstyle\pm 0.10$	93.09 $\scriptscriptstyle\pm 0.19$	24.95 $\scriptscriptstyle\pm 1.26$	40.02 $\scriptscriptstyle\pm 2.92$

Table 3: Backbone ablation study (mean

\pm

std across runs).

A

= augmentation;

I

= image-space loss. Both backbones have

\sim

12M parameters and the same network depth for a fair comparison. Best results are in bold .

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Mandibular reconstruction with vascularized bone grafts is complicated by donor-host nonunion, and current virtual surgical planning produces a geometric plan rather than a configuration that explicitly promotes bone union. We present OsteoOpt++, an image-to-decision planning loop for patient-specific mandibular reconstruction. A pre-operative computed tomography (CT) is converted into a personalized digital twin through template-to-patient registration and CT-derived updates of the muscle and temporomandibular-joint parameters. Bayesian optimization with an expected-improvement-plus acquisition rule then searches six clinically controllable cut-plane and donor-positioning variables under an apposition-driven objective and a safety-factor-regularized variant. The workflow was evaluated on three generic defects (body, symphysis, and ramus-body) and a total of 3+1 patient-specific cases, with 3 used for optimization and 1 for validation. In the generic cases, against a common surgical approach, cycle-averaged donor-mandible apposition increased by up to 29 percentage points (329% relative); in the patient-specific cases, against the surgeon-implemented day-5 post-operative configuration, by up to 26 percentage points. A 10% sensitivity analysis over eleven modeling parameters capped the change in the apposition-driven objective at 3% for generic cases and 4% for patient-specific cases, and the longitudinal case showed Dice overlap of 0.70 and 0.76 between predicted apposition and year-1 bone formation. Clinically, this provides surgeons with a pre-operative, image-driven recommendation for cut-plane orientation and donor placement that is predicted to improve union conditions over the configurations currently delivered in the operating room. The optimization and patient-specific modeling code is open source at https://github.com/hamidreza-aftabi/OsteoOpt.

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