### Patient population

We retrospectively analysed patients from a previously published study on the effects of anti-angiogenic therapy on the vascularity of HCC [3]. The study was approved by the Ethics Committee of ASST Monza–Ospedale San Gerardo (Monza, Italy) and all patients provided written informed consent.

Patients were enrolled between March 2012 and October 2016. Inclusion criteria were (1) a diagnosis of HCC; (2) Child–Pugh class A; (3) Eastern Cooperative Oncology Group performance status 0–1; (4) not having received previous systemic treatment for HCC; (5) no contraindications to CT imaging. Exclusion criteria were (1) Child–Pugh class B and C; (2) previous administration of *c-met* inhibitors; (3) concomitant radiotherapy; (4) a history of other malignancies or their concomitant presence; (5) presence of esophageal varices bleeding or of coagulation disorders; (6) glomerular filtration rate below 30 mL/min.

### CT protocol and scanning parameters

Patients underwent a clinical CT study before and after weight-tailored intravenous administration of an iodinated contrast agent at a 4.5 mL/s flow rate, using a 18 gauge catheter positioned into an antecubital vein. Arterial, portal venous, and equilibrium phases were acquired with a 2 mm collimation (pitch of 0.83). The bolus tracking technique was used to set individual acquisition times for the dynamic phases (*i.e.*, arterial, portal venous, and delayed phases). Images from this part of the original study were not used for the analysis reported in the present manuscript.

To avoid influence of previously administered contrast agent, the perfusion CT study was performed about 45 min afterwards. Perfusion studies were performed as follows: a 50 mL bolus of iodinated contrast (Xenetix 350; Guerbet, Aulnay, France) with a 350 mgI/mL concentration was injected at a 5 mL/s flow rate, acquiring 40 CT scan volumes on a 256-slice multi-detector-row (slab thickness 80 mm) at time intervals of 1.5 s. All patients were imaged on the same 256-slice CT scanner (Brilliance, iCT, Philips Medical Systems, Eindhoven, The Netherlands). Imaging parameters were: 100 kVp, 100 mAs, 512 × 512 matrix, slice thickness 2.5 mm, acquisition time 1.4 s. The acquisition began after a 5 s delay from intravenous contrast agent injection. A strap compressing the abdomen and limiting respiratory excursions was used to reduce respiratory motion artifacts.

### Image analysis

Image analysis was performed with a custom-made software (www.softefilm.eu). A trained radiologist (M.C., with 6 years of experience in abdominal imaging) pre-processed perfusion images by drawing four ROIs. The first ROI was drawn on the abdominal aorta between the emergence of the superior mesenteric artery and of the renal arteries: as in the reference study and the original manuscript by Miles [9], this ROI was used to define the feeding vessel time-enhancement curve, assuming that the same curve is conserved in the branching vessels. The second ROI was drawn on a HCC focus without evident necrosis, first selected on the clinical CT scans—because of their higher SNR and of the availability of a delayed acquisition—and afterwards identified and segmented on the perfusion CT images. The last two ROIs were drawn on the cortex of right kidney and on the biggest pancreatic region included in the study volume visible (as exemplified in Fig. 1). Afterwards, the aforementioned custom-made software translated each ROI on the same spatial coordinates of the remaining 39 frames. No attempt was made to spatially register the frames: this fact, combined with image noise, engendered a reduction of the whole quality of time-enhancement curves.

For image quality analysis, we computed the SNR as mean of the pre-contrast aortic HU value divided by the ROI standard deviation. To compute contrast-to-noise ratio (CNR), a fifth ROI was drawn on the paravertebral musculature in order to obtain:

$$\mathrm{CNR}=\frac{\mathrm{Aortic\;mean\;HU\;}-\mathrm{\;muscle\;mean\;HU}}{\mathrm{Muscle\;HU\;standard\;deviation}}$$

For each ROI, mean HU values with their standard deviation were computed alongside median HU values with their interquartile range. Then, the median HU values were used to compute the time-enhancement curve in the abdominal aorta with the Madsen formulation of the gamma variate function [11], as described below, obtaining the peak and α values. Conversely, for ROIs on HCC, kidney cortex, and pancreas, we computed time-enhancement curves from ROI median values with two different methods: (i) using a spline function (in order to remove image noise and obtain smoothed curves); (ii) by fitting a gamma variate function according to the Madsen formulation (to obtain peak tissue enhancement and *t*_{max}) [11].

### Perfusion evaluation

Using the whole set of time-enhancement data, the maximum slope method was used to computed perfusion two times: (i) using maximum slope values obtained from the spline function; (ii) using maximum slope values of the function determined as the maximal value of the first derivative of the gamma variate equation (using peak tissue enhancement, *t*_{max}, and aortic α value).

### Relationship between perfusion ratios and enhancement ratios

We computed for each patient the ratio between kidney/pancreas, kidney/HCC and pancreas/HCC perfusion and peak tissue enhancements. Perfusion data refer to the perfusions computed as maximal upslope of the time-enhancement curve obtained using all data points smoothed with the spline function.

### Mathematical model

Given that arterial circulation is made of terminal branching vessels, and all vessels in a definite anatomical region have the same circulation time, perfusion \(\left(\frac{\displaystyle\frac{ml}s}{ml_{tissue}}\right)\) can be measured as proposed by Peters et al. [8] and Miles [9]:

$$\mathrm{Perfusion}=\frac{\mathrm{Maximum\;slope\;of\;tissue\;enhancement}}{\mathrm{Peak\;feeding\;vessel\;enhancement}}$$

As previously mentioned, we aim to compute the maximum slope of tissue enhancement using only two single-energy CT scans (baseline and peak). Peak feeding vessel enhancement in the clinical setting can be inferred using the bolus test technique [12].

Vessel and tissue enhancement after the injection of a contrast bolus can be described with a gamma variate function [11, 13]. We use here the simplified formulation proposed by Madsen [11]:

\(\mathrm{Enhancement}={y}_{\mathrm{max}}\times {t}^{\alpha }\times {e}^{\alpha \left(1-t\right)}\) where *y*_{max} represents the maximum enhancement, *t* is defined as \(\mathrm{time}\text{/}{t}_{\mathrm{max}}\) (t_{max} being the time at which the function reaches its maximum), and α describes the bolus shape.

The bolus shape is conserved between arterial input and tissue time-enhancement curves [8, 9]: this allows to determine bolus shape from the arterial input curve (α) and use it together with the measured peak tissue enhancement to compute the tissue time-enhancement curve.

The maximal slope of the curve can be defined as the value of the first derivative of the Madsen equation at the time where the second derivative is 0. The first derivative is

$$dy={y}_{\mathrm{max}}\times \left(\left(\alpha \times {t}^{\alpha -1}\times {e}^{\alpha \left(1-t\right)}\right)-\left(\alpha \times {t}^{\alpha }\times {e}^{\alpha \left(1-t\right)}\right)\right)$$

the second derivative is

$$ddy={y}_{\mathrm{max}}\times \alpha \times {e}^{\alpha \left(1-t\right)}\times {t}^{\alpha -2}\times \left(\alpha \times {t}^{2}-2\times \alpha \times t+\alpha -1\right)$$

and the *t* at which the 0 of the second derivate is

$$t=\frac{\left(\alpha \pm \sqrt{\alpha }\right)}{\alpha }$$

Consequently, we can define a parameter *k* so that:

\(k=\left(\left(\alpha \times {t}^{\alpha -1}\times {e}^{\alpha \left(1-t\right)}\right)-\left(\alpha \times {t}^{\alpha }\times {e}^{\alpha \left(1-t\right)}\right)\right)\) with \(t=\frac{\left(\alpha \pm \sqrt{\alpha }\right)}{\alpha }\)

and finally:

$$\mathrm{Maximum\;slope\;of\;tissue\;enhancement}={y}_{\mathrm{max}}\times \frac{k}{{t}_{\mathrm{max}}}$$

### Relative perfusions

Considering two different regions of interest (ROIs) we can re-write:

$$\frac{{\mathrm{Organ\;blood\;flow}}_{{\mathrm{ROI}}_{1}}}{{\mathrm{Organ\;blood\;flow}}_{{\mathrm{ROI}}_{2}}}=\frac{\frac{{\mathrm{Maximum\;slope\;of\;tissue\;enhancement}}_{{\mathrm{ROI}}_{1}}}{{\mathrm{Peak\;feeding\;vessel\;enhancement}}_{{\mathrm{ROI}}_{1}}}}{\frac{{\mathrm{Maximum\;slope\;of\;tissue\;enhancement}}_{{\mathrm{ROI}}_{2}}}{{\mathrm{Peak\;feeding\;vessel\;enhancement}}_{{\mathrm{ROI}}_{2}}}}$$

Mathematically, the peak enhancement of the feeding vessel can be removed as:

$$\frac{{\mathrm{Organ\;blood\;flow}}_{{\mathrm{ROI}}_{1}}}{{\mathrm{Organ\;blood\;flow}}_{{\mathrm{ROI}}_{2}}}=\frac{{\mathrm{Maximum\;slope\;of\;tissue\;enhancement}}_{{\mathrm{ROI}}_{1}}}{{\mathrm{Maximum\;slope\;of\;tissue\;enhancement}}_{{\mathrm{ROI}}_{2}}}$$

Then, since

$$\frac{{\mathrm{Maximum\;slope\;of\;tissue\;enhancement}}_{{\mathrm{ROI}}_{1}}}{{\mathrm{Maximum\;slope\;of\;tissue\;enhancement}}_{{\mathrm{ROI}}_{2}}}=\frac{{y}_{{\mathrm{max}}_{\mathrm{ROI}1}}\times \frac{k}{{t}_{\mathrm{max}}}}{{y}_{{\mathrm{max}}_{\mathrm{ROI}2}}\times \frac{k}{{t}_{\mathrm{max}}}}$$

where

$$k=\left(\left(\alpha \times {t}^{\alpha -1}\times {e}^{\alpha \left(1-t\right)}\right)-\left(\alpha \times {t}^{\alpha }\times {e}^{\alpha \left(1-t\right)}\right)\right)$$

with \(t=\frac{\left(\alpha \pm \sqrt{\alpha }\right)}{\alpha }\), we can define

$$\frac{{\mathrm{Organ\;blood\;flow}}_{{\mathrm{ROI}}_{1}}}{{\mathrm{Organ\;blood\;flow}}_{{\mathrm{ROI}}_{2}}}=\frac{{y}_{{\mathrm{max}}_{{\mathrm{ROI}}_{1}}}}{{y}_{{\mathrm{max}}_{{\mathrm{ROI}}_{2}}}}$$

Consequently, the ratio of arterial phase tissue enhancement corresponds to the ratio of arterial perfusions.

### Statistical analysis

The relationship between the two perfusion computation methods and the relationship between perfusion ratios and enhancement ratios were investigated with linear regression and Bland–Altman analysis.