FREDERICK HOLMER FANN JONAS STRÖM SEEZ

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DEGREE PROJECT IN TECHNOLOGY, FIRST CYCLE, 15 CREDITS

STOCKHOLM, SWEDEN 2020

Development of an MRI-compatible

Multi-compartment Phantom for

Dynamic Studies

FREDERICK HOLMER FANN

JONAS STRÖM SEEZ

KTH ROYAL INSTITUTE OF TECHNOLOGY

SCHOOL OF ENGINEERING SCIENCES IN CHEMISTRY, BIOTECHNOLOGY AND HEALTH

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i

This project was performed in collaboration with

Department of Nuclear Medicine, Karolinska University Hospital

Supervisor at Department of Nuclear Medicine, Karolinska University Hospital: Alejandro

Sanchez-Crespo, PhD, Associate Professor

Development of an MRI-compatible Multi-

compartment Phantom for Dynamic Studies

Utveckling av MRI-kompatibel flerkammarfantom för

dynamiska studier

F R E D E R I C K H O L M E R F A N N J O N A S S T R Ö M S E E Z

Degree project in medical engineering First level, 15 hp

Supervisor at KTH: Tobias Nyberg, Mattias Mårtensson Examiner: Mats Nilsson

KTH Royal Institute of Technology

School of Engineering Sciences in Chemistry, Biotechnology and Health SE-141 86 Flemingsberg, Sweden

http://www.kth.se/cbh

2020

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Abstract

Medical imaging based on radioactive tracers exposes the patient to radiation. For this reason, a

phantom is preferably used for non-clinical studies such as routine quality assurance and

research. The aim of this project was to design, build and test a multi-compartment phantom to be

used in dynamic SPECT/CT, PET/CT and PET/MRI studies. By treating each compartment as a

biological system and plotting activity distribution, desired characteristics of the phantom can be

obtained. A software program was created to simulate compartment activity distribution for

different input parameters. Such parameters include number of compartments, administered

activity, flow rates between compartments and compartment volume. Based on the simulation,

the phantom was designed to meet the desired characteristics. Due to the outbreak of the SARS-

CoV-2 virus, no phantom could be built nor tested. Consequently, leading the project to create a

foundation that facilitates future building of the phantom.

Keywords: Dynamic studies, dynamic phantom, multi-compartment analysis, MRI compatible,

PET, SPECT, gamma camera, nuclear medicine.

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Sammanfattning

Medicinsk avbildning med radioaktiva spårämnen utsätter patienter för en stråldos. Av detta skäl

används företrädesvis en fantom för icke-kliniska studier såsom rutinmässig kvalitetssäkring och

forskning. Syftet med detta projekt var att designa, bygga och testa ett flerkammarfantom som

ska användas i dynamiska SPECT/CT, PET/CT och PET/MRI studier. Genom att behandla varje

kammare som ett biologiskt system och plotta aktivitetsfördelning kan önskade egenskaper hos

fantomen erhållas. Ett program skapades för att simulera aktivitetsdistributionen i

flerkammarfantomer för olika in parametrar så som antal kammare, administrerad aktivitet,

flöden mellan kammare och kammarvolym. Baserat på simuleringen utformades fantomen för att

uppfylla de önskade egenskaperna. På grund av utbrottet av SARS-CoV-2 viruset kunde ingen

fantom byggas eller testas. Följaktligen leddes projektet till att skapa en grund som underlättar

framtida byggande av fantomen.

Nyckelord: Dynamiska studier, dynamisk fantom, multi-kammar analys, MRI kompatibel, PET,

SPECT, gammakamera, nuklearmedicin

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iv

1. Introduction 1

1.1 Aim 1

1.2 Limitation 1

2. Background 2

2.1 Imaging Phantom 2

2.2 Imaging 3

2.3 Physical properties Fel! Bokmärket är inte definierat.

2.4 Radiotracer 4

3. Method 5

3.1 Software 5

3.2 Simulation 6

3.3 Phantom design 6

4. Results 7

4.1 Software 7

4.2 Simulation 10

4.3 Phantom design 12

5. Discussion 15

5.1 Software and Simulation 15

5.2 Phantom design 16

5.3 Future directions 16

6. Conclusion 17

7. References 18

Appendices

Appendix 1: Ordinary Differential Equations for multi-compartment analysis

Appendix 2: Multi-compartment setups

Appendix 3: Concept images of phantom

Appendix 4: Software code

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Abbreviations

A Activity with relating compartment index

V Volume with relating compartment index

Q Flow with relating compartment index. First index: from

compartment. Second index: to compartment

PET Positron Emission Tomography

SPECT Single Photon Emission Computed Tomography

MRI Magnetic Resonance Imaging

CT Computed Tomography

ODE Ordinary Differential Equations

FOV Field of view

SR Spatial Resolution

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1 1. Introduction

Dynamic studies using radiotracers injected into the patient and imaged with gamma or positron

cameras provide important information about internal organ/tumour functionality and disease

progression [1]. By measuring the variation of the radiopharmaceutical concentration in the

patient using gamma camera, Single Photon Emission Computed Tomography (SPECT) or

Positron Emission Tomography (PET), the relevant uptake and excretion rates can be calculated

using mathematical models. This is called multiple compartment analysis [2-4]. This method is

currently used in the clinical routine work at the Department of Nuclear Medicine, Karolinska

University Hospital. In addition to clinical use on patients, imaging devices such as Gamma

camera and PET imaging are also used for routine quality assurance and for research purposes.

Uncertainty and limitations in dynamic studies can be determined by administered activity,

acquisition time and camera specific physical properties. They affect the results of the studies and

determination of organ/tumour specific kinetic factors. For quality assurance and research these

are some parameters of interest [5-7].

In nuclear medicine, radiotracers are widely used for dynamic studies in patients. By injecting

radiotracers into a person, a dose of radiation is delivered. The dose delivered from one PET scan

is not insignificant [8]. If the scan is clinically justified, the crucial information provided is of

higher importance. The dose delivered to patients should always be minimized since exposure to

radiation has many negative effects, including developing cancer [9]. Therefore, performing

scans on people for quality assurance and for research purposes is unwanted.

Hence it would be desirable to have access to a dynamic phantom consisting of multiple

compartments to be placed in the camera instead. By treating each relevant tissue in context as a

compartment, the dynamic multi-compartment phantom can be used for non-clinical uses.

1.1 Aim

The aim of this project was to design, build and test a multi-compartment phantom that can

simulate human characteristics to be used for quality assurance and research in dynamic

SPECT/CT, PET/CT and PET/MRI studies.

1.2 Limitation

Simulating human characteristics and all its features is complex. Some features such as breathing

movement and anatomically correct structures were not included in the project.

The software simulations assume that the content of the compartments is homogeneous. Thereby

the calculation was simplified and did not include fluid dynamics.

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2 2. Background

2.1 Imaging Phantom

A phantom is a specially designed object that mimics a desirable behaviour. In medical imaging

phantoms are used to analyse, evaluate and tune medical imaging devices. This is preferable due

to its consistent behaviour and avoids risking living subjects.

Using compartments to analyse distribution of pharmaceuticals or radiotracers in vivo is a

method called multi-compartment analysis. In this context, compartments refer to regions of

interests such as organs or parts of an organ. For instance, a heart can be associated with a four-

compartment model, each compartment representing one of its four distinct chambers. Multi-

compartment analysis is a clinically used mathematical model. The reason to use this model is

that its associated mathematical functions describe the observed data very well and are thus

practical to use [3, 4, 10].

Phantoms can be built in a variety of constellations due to the amount of compartment models

that can be represented (Figure 1). They can be static or dynamic, both can include anatomical

structures while dynamic phantoms also can simulate pharmaceutical distribution and organ

functionality.

Figure 1: Example of a multi-compartment setup of a biological system.

An earlier model of a dynamic multi-compartment

phantom has been built and used at the Department

of Nuclear Medicine at Karolinska University

Hospital (Figure 2). The phantom contains around 30

litres of water making it cumbersome to manage.

Because of its size it fits only in a gamma camera. It

is built of acrylic plates that are screwed and glued

together and leakage has started to occur.

Figure 2: Multicompartment phantom used at the Department of Nuclear Medicine at Karolinska

University Hospital

.

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3 2.2 Imaging

Medical imaging is used to visualize the internals of the body to get information of its structure

and functionality. Functional imaging is taken over a timespan to receive dynamic information

from the patient. Functional imaging such as SPECT, PET and gamma cameras use radioactive

tracers. When the tracer spreads in a patient the distribution of the tracer is imaged. Different

types of radiotracers can be used depending on the tissue of interest. Such type of radiotracers has

the property to accumulate at a specific target and are called targeting radiotracers [1].

Complement to functional imaging is Computed Tomography (CT) and Magnetic Resonance

Imaging (MRI) which provide structural information of the anatomy. A combination of the two

different styles of imaging is called hybrid imaging and is widely used today [11]. CT and MRI

can include PET and SPECT equipment mounted in the camera gantry, allowing for sequential

acquisition of functional and anatomical images. Commonly used combinations are SPECT/CT,

PET/CT and PET/MRI [11, 12]. Hybrid systems enable functional images to be merged with the

corresponding anatomical correlate. Images can therefore be adjusted after anatomic structures

and in that way give more accurate information to the user [11]. The Department of Nuclear

Medicine at Karolinska University Hospital comprises several of these hybrid systems, including

a PET/MRI with a 3 Tesla magnetic field.

When imaging a patient, factors such as administered activity, acquisition time and camera-

specific physical properties are entities to consider. The amount of administered activity is a

balance between the dose delivered to the patient and Signal to Noise Ratio (SNR). Acquisition

time is the time that it takes to capture one frame. Long acquisition time gives a better SNR

however, it restricts the number of frames acquired.

Dynamic studies are based on a combination of different parameters. Uncertainty and limitations

in dynamic studies can be determined by the factors mentioned in the earlier paragraph. This will

affect the results of the studies and determination of organ or tumour specific kinetic factors such

as movements of nearby tissues and fluids. For quality assurance and research these are some

parameters of interest.

2.3 Ferromagnetism

For a phantom to work in an MRI, components in the vicinity of the MRIs magnetic field must be

composed of non-ferromagnetic materials. Atoms within a material produce a magnetic dipole

moment. This magnetic dipole moment is generated from charged particles spinning or orbiting

around the atom. Direction of the magnetic dipole moment varies for each atom. Depending on

the vector sum the dipole moment can either cancel out or add up. In a ferromagnetic material

more of the atom's dipole moment points in a net direction and therefore the material has a

magnetic field. This is called spontaneous magnetization. However, if the vector sum becomes

zero or very small the material is non-ferromagnetic and has no magnetic field. Common non-

ferromagnetic metals are aluminium and copper. A non-ferromagnetic material can become

magnetic if it is exposed to an external magnetic field. This aligns more of the atom's magnetic

dipole moment to point in the same direction as the external magnetic field [13].

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4 2.4 Radiotracer

A radiotracer is a chemical compound where some of the atoms have been replaced by a

radionuclide. A radionuclide is unstable by emitting excessive nuclear energy. This can be done

through gamma radiation, internal conversion or by creating and emitting an alpha or beta

particle. There are two types of beta particles, beta negative and beta positive. They have

respectively negatively and positively charged particles also known as electrons and positrons [1,

14].

The radiotracer used in PET studies is labelled with a positron emitting radionuclide. Such a

radionuclide decays by emitting a positron which then almost immediately results in annihilation

with an electron emitting two gamma rays in opposite directions [15].

Particles/energy emitted in either form mentioned above are harmful to the human body.

Absorbed dose is a measurement indicating the amount of radiation absorbed in relation to the

mass. It assesses the potential for biochemical changes in living tissue and is measured in Gray

(Gy) which is equivalent to J/kg. Equivalent dose includes the type of radiation by multiplying

absorbed dose with a weight factor depending on the type of radiation and is measured in Sievert

(Sv). Effective dose has in additional weight factor for the type of tissue and assesses the potential

for long term effects.

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5 3. Method

This project started with development of a software program. The program was used to simulate

the dynamic properties of the phantom. Results from the simulations established structural

specifications which were used to design the phantom. Due to the outbreak of the SARS-CoV-2

virus, building and testing the phantom were unaccomplished. The initial plan was to follow the

flowchart (Figure 3).

Figure 3: The initial planned project workflow, showcasing an interrupt after “Phantom Design”

3.1 Software

The program generates and solves Ordinary Differential Equations (ODE) describing the activity

concentration for each compartment. Relevant parameters to consider during the software

development were:

- Number of compartments

- Flow rates

- Compartment volumes

- Initial conditions regarding the activity distribution in the compartment

- Type of isotope

The software was created in Matlab App Designer 9.8 (MathWorks Inc., Natick, MA, USA). The

development of the program started by writing code that generates mathematical expression for

the activity distribution. The mathematical expression consisted of a system of ODEs and solved

using matlabs dsolve function. The solution contains functions of activity concentration for each

compartment. These functions were plotted using matlabs fplot function.

The collected data of activity distribution from an old phantom imaged with a gamma camera

was used to verify that the software was working correctly. It had a setup of two compartments

(Appendix 2: Multi-compartment setups). The parameters used in the test were simulated. Graphs

of activity distribution functions from the simulation and acquired data were compared.

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6 3.2 Simulation

Before designing the physical phantom, simulations were made to assure its dynamic properties.

The software program was used to simulate compartment activity distribution for different input

parameters until a desired plot was obtained. Important properties of the plot were that the time

frame of relevant data gathering is practically implementable and that an intersection of

equilibrium existed within that time frame. Limitations had to be considered when adjusting the

parameters since they are to be implemented in a real phantom. Flow rates must be driven and

compartment sizes cannot be too big. The total volume plot (Figure 10) was used to find the

volume required by the phantom within the desired time frame.

3.3 Phantom design

Important characteristics to consider during the development of the phantom design were:

● Leakproof

- Minimizing structural weaknesses to prevent leakage when exposed to structural

stress and pressure.

● Size

- Size limitation for the imaging devices was a cylindrical hole with diameter of 60

cm and a depth of 20 cm excluding the examination table.

● Dynamical properties

- Adjustable flow between compartments.

- Activity distribution correlates in the same magnitude as clinical examinations.

● MRI compatible

- Ferromagnetic materials could not be in the vicinity of the MRIs magnetic field.

● Accessibility to tools

Compartment volume and flow rates were obtained from the simulation and used to design a

concept of the phantom. This was done in the 3D program Blender 2.82 (Blender Foundation,

Amsterdam, Netherlands). Models of all the compartments were made and the dimensions were

selected to fit the result of the simulation and camera restrictions.

To give the phantom its dynamical properties, a concept of an MRI-compatible pump was

developed. It consisted of a pneumatic stepper engine and two peristaltic pumps adapted to be 3D

printed. The two objects were found on an open source website Thingiverse

(www.thingiverse.com). To finalize the phantom concept, the two objects were added to the other

models in Blender. A scene in Blender was created and concept pictures were generated from it.

The materials necessary for the design of the phantom were listed.

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7 4. Results

4.1 Software

A software was created that with its main functionality is to generate, solve and plot the solutions

of the ODE system for a multi-compartment setup (Appendix 1: ODE for multi-compartment

analysis, Appendix 4: Software code). The software can simulate the behaviour of a variety of

multi-compartment systems with different numbers of compartments, volumes and flows. It has a

graphical user interface displaying input parameters as well as plotting activity distribution and

volume requirements (Figure 4). The image compartments 1 and 2 are assumed to always have

constant volume by having water supplied from the supply compartment in and out (Appendix 2:

Multi-compartment setups).

Figure 4: Main window of the software. Graphical display of “Activity Distribution”, “Total volume” and the input parameters.

Activity distribution and total volume are displayed in two separate graphs. The two graphs are

generated from input parameters when the “Plot” button is pressed. A toggle switch in the bottom

left of the activity distribution graph switches between displaying activity [kBq] or activity

concentration [kBq/L] of each compartment.

The user can choose the number of compartments. The UI

is limited to a system of up to four compartments (Figure

5). Choosing different system setups changes the main

window by adding or removing input fields.

Figure SEQ Figure \* ARABIC 4

Figure 5: Drop down menu of selectable compartment setup.

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8 The input parameters are categorized into three sections, “Activity”, “Flow” and “Volume”. Each

section displays their input parameters respectively. The input fields have drop-down menus for

different units (Figure 4).

Under the “Activity” section there is a drop-down menu where the user can choose among

predefined isotopes or add a new. The software can simulate two different types of activity

infusions, bolus and continuous. The buttons “Bolus” and “Continuous” change the interface

(Figure 4 and 6). A bolus injection is defined as an instantaneous infused activity at time zero

while a continuous infusion is defined as a constant activity infusion over a defined time interval.

All activity parameters like type of dose infusion and type of isotope were based on clinical data

[16].

The button “Show Data” (Figure 4) button opens another window where more detailed data of the

simulation is shown in a table (Figure 7). The equations of the activity distribution are also

displayed. The window has a button “Export to excel”, allowing the user to export the data table

to an excel file.

Figure 7: Window of system data appearing when “Show Data” button is pressed in the main window.

Displaying data in a table of activity, activity concentration and volume (left), a list of associated equations (right). The button “Export data” (bottom right) exports the data to an excel sheet.

Figure 6: View of bolus and continuous activity input fields in the main windows activity section. The interface is updated depending on the selected buttons, bolus or continuous button.

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9 Comparison between the collected data of activity distribution generated from the gamma camera

(Figure 8) and its correlated simulation plot provided by the software (Figure 9).

Figure 8: Graphical display of a two-compartment phantom used at

Karolinska University Hospital. Data is obtained with a gamma camera over a 35 minute period and an acquisition time of 20 seconds.

Figure 9: Graphical display of the software’s simulation of a two-compartment phantom used at Karolinska University Hospital.

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10 4.2 Simulation

The selected simulation data used in the software to simulate properties of the final multi-

compartment phantom.

Table 1: Value of input parameters for the selected simulation.

Isotope

Tc99-m (half-life 6h)

Initial activity (Bolus)

Image compartment 1 10 MBq

Image compartment 2 0 Bq

Volume

Compartment 1 and 2 0.5 litre

Compartment In 4.5 litre

Compartment Out 5.5 litre

Flow

Q12 150 ml/min

Q21 150 ml/min

Q1out 150 ml/min

Q2out 0 ml/min

Time

Sample time 30 minutes

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11 The simulated phantom properties provided from the multi-compartment analysis is described by

the following mathematical equations along with their graphical representation (Figure 10).

Equations

𝐴

₁

(𝑡) = 7.2 ∗ 10

⁶

∗ 𝑒

^{−0.013∗𝑡}

+ 2.8 ∗ 10

⁶

∗ 𝑒

^−1.9∗10⁻³^∗𝑡

𝐴

₂

(𝑡) = 4.5 ∗ 10

⁶

∗ 𝑒

^−1.9∗10⁻³^∗𝑡

− 4.5 ∗ 10

⁶

∗ 𝑒

^{−0.013∗𝑡}

𝐴

_𝑜𝑢𝑡

(𝑡) = 1.0 ∗ 10

⁷

∗ 𝑒

^−3.2∗10⁻⁵^∗𝑡

− 2.8 ∗ 10

⁶

∗ 𝑒

^{−0.013∗𝑡}

− 7.2 ∗ 10

⁶

∗ 𝑒

^−1.9∗10⁻³^∗𝑡

Plot

Figure 10: Simulation of the selected phantom setup, displaying its activity distribution equations 𝐴₁ and 𝐴₂ (top), and the total volume required plotted under the same period (bottom).

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12 4.3 Phantom design

The final phantom setup consists of two smaller imaging compartments and two larger supply

compartments (Figure 11). The two smaller compartments are those of interest to image and are

therefore positioned in the FOV of the camera. The two larger compartments are placed above

and under the two smaller compartments. Their purpose is to supply water to the two smaller

compartments to create dynamical characteristics.

Figure 11: Visual representation of the two-compartment setup.

The image compartments 1 and 2 consisted of a cylindrical shape, made of acrylic with threaded

lids of plastic (Figure 12). They have the same size with a diameter of 10 cm and a height of 6.4

cm resulting in a volume of 0.5 litre. The two larger compartments are cuboid and have a cross

sectional area of 20 cm x 30 cm. The larger compartment placed on top has a height of 7.5 cm

resulting in a volume of 4.5 litre. The compartment placed on the bottom has a height of 10 cm,

resulting in a volume of 6 litre. PVC hoses of 8 mm connect the compartments to each other. The

connection between hose and compartment are easily removed to allow for access. The two

smaller compartments also have injections sites via the connections to allow for activity to be

injected. These sites also have a way to remove air from the system.

Figure 12: 3D design model created in Blender of the two-compartment setup. Front view (left), side view (right).

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13 Flow between the larger compartments and smaller compartments is driven by gravitational force

and is regulated via valves. Flow between the two smaller compartments must be driven. An

option for this is the use of a non-ferromagnetic pump consisting of a pneumatic stepper engine

(www.thingiverse.com/thing:2813533 Created by user: Vincent Groenhuis) and two peristaltic

pumps (www.thingiverse.com/thing:317726 Created by user: rzweije). Open Source files of

models that could be 3D printed were found at Thingiverse. This option requires the pneumatic

stepper engine to have access to an air supply, controllable valves and a controller. The valves are

four 5V solenoid valves that are connected to a microcontroller board, e.g. Arduino uno

(Arduino, Sommerville, MA, USA), regulating opening and closing of the valves. Each solenoid

valve is connected via a 10 m hose adapted for pressure to individual chambers in the pneumatic

stepper engine. This allows the user to operate the pneumatic stepper engine at a safe distance

from radioactivity and magnetic field. By controlling the inlet of air to each chamber pistons are

moved that rotates a shaft which drive the two peristaltic pumps (Figure 13).

Figure 13: 3D models created in Blender. Upper left: front view of pneumatic stepper engine (left) and peristaltic pump (right). Upper right: rear view of pneumatic stepper engine (right) and peristaltic pump (left). Bottom left: open front view of pneumatic stepper engine (left) and peristaltic pump (right).

of the pneumatic stepper engine and the peristaltic pump.

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14 In the final design, the in compartment and compartment 1 and 2 are placed in a placeholder to be

put in the camera gantry. The out compartment is put outside the camera at a lower level (Figure

14). More detailed pictures of the phantom is located in Appendix 3: Concept images of the

phantom. Materials required for this phantom are seen in Table 2.

Table 2: The materials needed for this phantom.

Item Amount

Small compartment 2 pieces

Large compartment 2 pieces

PVC hose 8 mm 4 m

Threatened connection point 10 pieces

Shut off valve 4 pieces

Placeholder 1 piece

Pneumatic stepper engine 1 piece

Air compressor 1 piece

Pressure resistive hose 40 m

Microcontroller board 1 piece

5V solenoid air valve 4 pieces

Peristaltic pump 2 pieces

Silicone hose 8 mm 0.5 m

Figure 14: Final conceptual phantom design. Compartment in and compartment 1 and 2 are put in a placeholder that are intended to be placed in the imaging camera (back). Compartment out is intended to be placed outside the camera at a lower level.

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15 5. Discussion

Dynamic studies using a multi-compartment phantom can be used for routine quality assurance

and research. A phantom is preferable due to its consistent behaviour and not risking living

subjects. It can mimic desired behaviour and analyse distribution of pharmaceuticals or

radiotracers in vivo. The software is based on multi-compartment analysis to simulate activity

distribution in a multi-compartment phantom.

As the project progressed, circumstances were changed due to the outbreak of the SARS-CoV-2

virus. Restrictions in the community limited access to the workshop and delivery time of

materials became uncertain. Because of this, the aim to build a workable phantom was not

achieved. Instead, more preparatory work was done to facilitate future building.

5.1 Software and Simulation

The essence of having a software program is to solve the ODE system describing the dynamics of

the phantom for arbitrary input parameters. Since the project changed direction, more focus was

put on developing a software program with more functionality. To extend simulation usability,

the following parameters had to be adjustable (Figure 4 - 6).

- Number of compartments

- Flow rates

- Compartment volumes

- Initial conditions regarding the activity distribution in the compartments

- Type of isotope

These parameters affect the phantoms properties by regulating the activity distribution in the

compartments. They are needed to be considered when determining the design.

A high number of compartments increases adjustability and complexity of a simulation and in

that way dictating what can be simulated. In clinical relation this can be described as interactions

between biological systems (Figure 1) [3]. However, when designing a phantom, increasing the

number of compartments results in increased size and water supply needed. Due to this, a two-

compartment setup was chosen (Figure 11).

Activity distribution in each compartment can be controlled to mimic desirable characteristics by

regulating flows rates. Outflows are directly related to the amount of water supply necessary in

the system, thus affecting the compartment sizes. Flow rate and volume parameters selected

(Table 1) provide an activity distribution plot for a duration of 30 minutes with a max required

volume of 5.5 litres (Figure 7 and 10), fulfilling the conditions listed in method.

Type of isotope and activity infusion also affects the activity distribution. These parameters can

vary in the clinic depending on the patient and type of examination [17]. Therefore, it is

beneficial to include them as parameters in the software (Figure 6).

The software can be used to design other phantoms with different setups. However, the UI will

not support more than four compartments (Figure 5). The code that manages the generation of

data is written generically and can theoretically handle an unlimited number of compartments.

The simulation assumes that the activity is distributed homogeneously within the compartment

which is not the case in reality. Without the ability to test how the activity is spreading, the effect

of this on the system is uncertain. The software was compared to an old phantom in use at

Karolinska University Hospital (Figure 8 and 9). For this particular comparison, the results show

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16 small differences with close to identical plots. This was only one test and no conclusions can be

drawn regarding the software’s accuracy. The simulation does not account for fluid dynamics

which is a possible source of error.

5.2 Phantom design

Parts where modelled in a 3D software so that interactions between them could be visualised.

Factors that do not directly affect the phantom properties were also considered. This includes

costs, delivery time of materials and accessibility of tools needed for construction.

The imaged compartments 1 and 2 are cylindrical shaped to reduce the critical area for possible

leakage. The associated lids are threaded to enable the compartments to be opened. When

determining the dimension of the compartments, Spatial Resolution (SR) are considered. SR is

dependent on how much signal is received. Uniform geometry is desirable to have equal SR in

every direction. The larger in/out compartments are exposed to less pressure and structural stress

and are therefore designed to maximise volume efficiency (Figure 12 and 14).

The project was meant to have a circular workflow. Testing the phantom as a whole should find

improvements that can be done, completing the feedback (Figure 3). To enable this, the phantom

was designed so that individual parts are easily replaceable. Because no phantom is built nor

tested, uncertainties in its functionality can first be determined when the phantom is built.

Therefore, disrupt in the workflow probably affected the final design (Figure 14, Table 2).

The pneumatic pump (Figure 13) is a component that has questionable functionality. The main

reason it is included in the final design is because it is MRI-compatible. It is also easily

accessible by being 3D printable and the flow rate is adjustable via a microcontroller board. An

alternative to the pneumatic stepper engine and the two peristaltic pumps is a piezoelectric pump

that is MRI-compatible. This option was discarded due to high costs and low power supply. If

MRI-compatibility is not an issue the pneumatic pump can be replaced with a regular water

pump.

5.3 Future directions

Further development of the phantom can be done by adding simulated breathing movement. The

two imaging compartments can be placed on an inflating and deflating air cushion controlled by

the same microcontroller board as the pneumatic pump. This could be implemented if movement

artefacts are wanted to be simulated.

Another development is to replace the simplified imaging compartments with already existing

static phantoms that is more anatomically correct. Implementing dynamics to statics phantoms

can lead to interesting projects.

The software allowed to theoretically simulate an unlimited number of compartments which gave

an idea to design a matrix compartment phantom. Such a phantom would consist of a matrix of

compartments, all being connected to each other. The generation of data required for this suits the

software well. With many compartments, a complex system could be simulated such as the whole

body or cellular interactions (Figure 1).

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17 6. Conclusion

This project provided a software and a conceptual dynamic multi-compartment phantom that is

MRI-compatible. Due to limitations, the final aim was not achieved. Although no physical

phantom was built nor tested, results provide a foundation for this to be achieved. The conceptual

phantom is adaptable to modifications by being designed so that individual parts are replaceable.

The software and conceptual phantom provided are applicable in different dynamic studies.

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18 7. References

[1] S. S. Gambhir, "Molecular imaging of cancer with positron emission tomography,"

Nature Reviews Cancer, vol. 2, no. 9, pp. 683-693, 2002/09/01 2002, doi:

10.1038/nrc882.

[2] D. R. Mould and R. N. Upton, "Basic concepts in population modeling, simulation, and

model-based drug development-part 2: introduction to pharmacokinetic modeling

methods," (in eng), CPT Pharmacometrics Syst Pharmacol, vol. 2, no. 4, pp. e38-e38,

2013, doi: 10.1038/psp.2013.14.

[3] M. H. Reid and H. B. Hechtman, "A multicompartment analysis of the lung," Medical

and biological engineering, vol. 12, no. 4, pp. 405-414, 1974/07/01 1974, doi:

10.1007/BF02478595.

[4] D. Mould and R. Upton, "Basic Concepts in Population Modeling, Simulation, and

Model-Based Drug Development—Part 2: Introduction to Pharmacokinetic Modeling

Methods," CPT Pharmacometrics Syst Pharmacol, vol. 2, no. 4, p. 38, 2013, doi:

10.1038/psp.2013.14.

[5] C. J. Scott et al., "Reduced acquisition time PET pharmacokinetic modelling using

simultaneous ASL-MRI: proof of concept," (in eng), J Cereb Blood Flow Metab, vol. 39,

no. 12, pp. 2419-2432, Dec 2019, doi: 10.1177/0271678x18797343.

[6] I. Goethals, Y. D'Asseler, A. Dobbeleir, K. Deblaere, and H. Ham, "The effect of

acquisition time on visual and semi-quantitative analysis of F-18 FDG-PET studies in

patients with head and neck cancer," (in eng), Nucl Med Commun, vol. 31, no. 3, pp. 227-

31, Mar 2010, doi: 10.1097/MNM.0b013e328334fbfd.

[7] C. J. Scott et al., "Short Acquisition Time PET/MR Pharmacokinetic Modelling Using

CNNs," in Medical Image Computing and Computer Assisted Intervention – MICCAI

2018, Cham, A. F. Frangi, J. A. Schnabel, C. Davatzikos, C. Alberola-López, and G.

Fichtinger, Eds., 2018// 2018: Springer International Publishing, pp. 48-56.

[8] A. Kaushik et al., "Estimation of radiation dose to patients from (18) FDG whole body

PET/CT investigations using dynamic PET scan protocol," (in eng), Indian J Med Res,

vol. 142, no. 6, pp. 721-31, Dec 2015, doi: 10.4103/0971-5916.174563.

[9] B. Huang, M. W. Law, and P. L. Khong, "Whole-body PET/CT scanning: estimation of

radiation dose and cancer risk," (in eng), Radiology, vol. 251, no. 1, pp. 166-74, Apr

2009, doi: 10.1148/radiol.2511081300.

[10] L. E. Gerlowski and R. K. Jain, "Physiologically Based Pharmacokinetic Modeling:

Principles and Applications," Journal of Pharmaceutical Sciences, vol. 72, no. 10, pp.

1103-1127, 1983, doi: 10.1002/jps.2600721003.

[11] D. W. Townsend, "Multimodality imaging of structure and function," (in eng), Phys Med

Biol, vol. 53, no. 4, pp. R1-r39, Feb 21 2008, doi: 10.1088/0031-9155/53/4/r01.

[12] H. Hricak et al., "Global Trends in Hybrid Imaging," Radiology, vol. 257, no. 2, pp. 498-

506, 2010, doi: 10.1148/radiol.10100579.

[13] Z. H. I. Sun, M. Guo, J. Vleugels, O. Van der Biest, and B. Blanpain, "Processing of non-

ferromagnetic materials in strong static magnetic field," Current Opinion in Solid State

and Materials Science, vol. 17, no. 4, pp. 193-201, 2013/08/01/ 2013, doi:

https://doi.org/10.1016/j.cossms.2013.05.001.

[14] "New physics for medical physics," Nature Reviews Physics, vol. 1, no. 9, pp. 523-523,

2019/09/01 2019, doi: 10.1038/s42254-019-0104-9.

[15] A. A. van der Veldt, E. F. Smit, and A. A. Lammertsma, "Positron Emission Tomography

as a Method for Measuring Drug Delivery to Tumors in vivo: The Example of

[(11)C]docetaxel," (in eng), Front Oncol, vol. 3, p. 208, 2013, doi:

10.3389/fonc.2013.00208.

(26)

19 [16] I. o. Medicine, Isotopes for Medicine and the Life Sciences. Washington, DC: The

National Academies Press (in English), 1995, p. 144.

[17] R. E. Snyder, T. R. Overton, D. P. Boisvert, and K. C. Petruk, "An automatic bolus

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Radiol, vol. 49, no. 588, pp. 1033-6, Dec 1976, doi: 10.1259/0007-1285-49-588-1033.

(27)

1 Appendix 1: Ordinary Differential Equations for multi-

compartment analysis

𝐴

_𝑖

: Activity intensity of compartment 𝑖

𝑄

_𝑖𝑗

: Flowrate from compartment 𝑖 to compartment 𝑗

𝑉

_𝑖

: Volume of compartment 𝑖

𝑅

_𝑖

: Activity rate of compartment 𝑖

𝜆: Decay constant of the isotope

𝐻: Heaviside function

One Compartment Setup:

Bolus

𝑑𝐴

₁

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₁

(𝑡) − 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡)

𝑑𝐴

_𝑜𝑢𝑡

(𝑡)

𝑑𝑡 = 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

_𝑜𝑢𝑡

(𝑡)

Continuous

𝑑𝐴

₁

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₁

(𝑡) − 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑅

₁

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

_𝑜𝑢𝑡

(𝑡)

𝑑𝑡 = 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

_𝑜𝑢𝑡

(𝑡)

Two Tank setup:

Bolus

𝑑𝐴

₁

(𝑡)

𝑑𝑡 = 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

₁

(𝑡) − 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡)

𝑑𝐴

₂

(𝑡)

𝑑𝑡 = 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

₂

(𝑡) − 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡)

𝑑𝐴

_𝑜𝑢𝑡

(𝑡)

𝑑𝑡 = 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

_𝑜𝑢𝑡

(𝑡) + 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡)

Continuous

𝑑𝐴

₁

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₁

(𝑡) − 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑅

₁

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

₂

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₂

(𝑡) + 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑅

₂

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

_𝑜𝑢𝑡

(𝑡)

𝑑𝑡 = 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

_𝑜𝑢𝑡

(𝑡) + 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡)

(28)

2 Three Tank setup:

Bolus

𝑑𝐴

₁

(𝑡)

𝑑𝑡 = 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝑄

₁₃

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

₁

(𝑡) + 𝑄

₃₁

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡)

𝑑𝐴

₂

(𝑡)

𝑑𝑡 = 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

₂

(𝑡) − 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₂₃

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

₃₂

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡)

𝑑𝐴

₃

(𝑡)

𝑑𝑡 = 𝑄

₁₃

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

₃

(𝑡) + 𝑄

₂₃

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₃₁

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

₃₂

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

_{3𝑜𝑢𝑡}

𝑉

₃

∗ 𝐴

₃

(𝑡)

𝑑𝐴

_𝑜𝑢𝑡

(𝑡)

𝑑𝑡 = 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

_𝑜𝑢𝑡

(𝑡) + 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

_{3𝑜𝑢𝑡}

𝑉

₃

∗ 𝐴

₃

(𝑡)

Continuous

𝑑𝐴

₁

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₁

(𝑡) − 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝑄

₁₃

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

₃₁

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑅

₁

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

₂

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₂

(𝑡) + 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₂₃

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

₃₂

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑅

₂

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

₃

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₃

(𝑡) + 𝑄

₁₃

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑄

₂₃

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₃₁

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

₃₂

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

_{3𝑜𝑢𝑡}

𝑉

₃

∗ 𝐴

₃

(𝑡) + 𝑅

₃

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

_𝑜𝑢𝑡

(𝑡)

𝑑𝑡 = (𝑄

_{1𝑜𝑢𝑡}

∗ 𝐴

₁

(𝑡))/𝑉

₁

− 𝜆 ∗ 𝐴

_𝑜𝑢𝑡

(𝑡) + 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

_{3𝑜𝑢𝑡}

𝑉

₃

∗ 𝐴

₃

(𝑡)

Four Tank setup:

Bolus

𝑑𝐴

₁

(𝑡)

𝑑𝑡 = 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝑄

₁₃

𝑉

₁

∗ 𝐴

₁

(𝑡) − (𝑄

₁₄

∗ 𝐴

₁

(𝑡))/𝑉

₁

− 𝜆 ∗ 𝐴

₁

(𝑡) + 𝑄

₃₁

𝑉

₃

∗ 𝐴

₃

(𝑡) + 𝑄

₄₁

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡)

𝑑𝐴

₂

(𝑡)

𝑑𝑡 = 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

₂

(𝑡) − 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₂₃

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₂₄

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

₃₂

𝑉

₃

∗ 𝐴

₃

(𝑡)

+ 𝑄

₄₂

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡)

𝑑𝐴

₃

(𝑡)

𝑑𝑡 = 𝑄

₁₃

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

₃

(𝑡) + 𝑄

₂₃

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₃₁

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

₃₂

𝑉

₃

∗ 𝐴

₃

(𝑡)

− 𝑄

₃₄

𝑉

₃

∗ 𝐴

₃

(𝑡) + 𝑄

₄₃

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

_{3𝑜𝑢𝑡}

𝑉

₃

∗ 𝐴

₃

(𝑡)

𝑑𝐴

₄

(𝑡)

𝑑𝑡 = 𝑄

₁₄

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

₄

(𝑡) + 𝑄

₂₄

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

₃₄

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

₄₁

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

₄₂

𝑉

₄

∗ 𝐴

₄

(𝑡)

− 𝑄

₄₃

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

_{4𝑜𝑢𝑡}

𝑉

₄

∗ 𝐴

₄

(𝑡)

(29)

3 𝑑𝐴

_𝑜𝑢𝑡

(𝑡)

𝑑𝑡 = 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

_𝑜𝑢𝑡

(𝑡) + 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

_{3𝑜𝑢𝑡}

𝑉

₃

∗ 𝐴

₃

(𝑡) + 𝑄

_{4𝑜𝑢𝑡}

𝑉

₄

∗ 𝐴

₄

(𝑡)

Continuous

𝑑𝐴

₁

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₁

(𝑡) − 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝑄

₁₃

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝑄

₁₄

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

₃₁

𝑉

₃

∗ 𝐴

₃

(𝑡)

+ 𝑄

₄₁

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑅

₁

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

₂

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₂

(𝑡) + 𝑄

₁₂

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝑄

₂₁

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₂₃

∗ 𝐴

₂

(𝑡)

𝑉

₂

− 𝑄

₂₄

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

₃₂

𝑉

₃

∗ 𝐴

₃

(𝑡)

+ 𝑄

₄₂

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑅

₂

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

₃

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₃

(𝑡) + 𝑄

₁₃

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑄

₂₃

𝑉

₂

∗ 𝐴

₂

(𝑡) − 𝑄

₃₁

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

₃₂

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

₃₄

𝑉

₃

∗ 𝐴

₃

(𝑡)

+ 𝑄

₄₃

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

_{3𝑜𝑢𝑡}

𝑉

₃

∗ 𝐴

₃

(𝑡) + 𝑅

₃

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

₄

(𝑡)

𝑑𝑡 = −𝜆 ∗ 𝐴

₄

(𝑡) + 𝑄

₁₄

𝑉

₁

∗ 𝐴

₁

(𝑡) + 𝑄

₂₄

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

₃₄

𝑉

₃

∗ 𝐴

₃

(𝑡) − 𝑄

₄₁

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

₄₂

𝑉

₄

∗ 𝐴

₄

(𝑡)

− 𝑄

₄₃

𝑉

₄

∗ 𝐴

₄

(𝑡) − 𝑄

_{4𝑜𝑢𝑡}

𝑉

₄

∗ 𝐴

₄

(𝑡) + 𝑅

₄

∗ (1 − 𝐻(𝑡 − 𝑡

_{𝑓𝑟𝑜𝑚}

) ∗ (𝐻(𝑡 − 𝑡

_𝑡𝑜

) − 1))

𝑑𝐴

_𝑜𝑢𝑡

(𝑡)

𝑑𝑡 = 𝑄

_{1𝑜𝑢𝑡}

𝑉

₁

∗ 𝐴

₁

(𝑡) − 𝜆 ∗ 𝐴

_𝑜𝑢𝑡

(𝑡) + 𝑄

_{2𝑜𝑢𝑡}

𝑉

₂

∗ 𝐴

₂

(𝑡) + 𝑄

_{3𝑜𝑢𝑡}

𝑉

₃

∗ 𝐴

₃

(𝑡) + 𝑄

_{4𝑜𝑢𝑡}

𝑉

₄

∗ 𝐴

₄

(𝑡)

(30)

1 Appendix 2: Multi-compartment setups

Visual representation of the one-, three-, four-compartment setup.

(31)

1 Appendix 3: Concept images of the phantom

Compartments

Pump

(32)

2

(33)

3 Placeholder

(34)

4

(35)

5 Concept Phantom Setup

(36)

1 Appendix 4: Software code

classdef Phantom_app < matlab.apps.AppBase

% Properties that correspond to app components properties (Access = public)

UIFigure matlab.ui.Figure

Activity_plot matlab.ui.control.UIAxes PlotButton matlab.ui.control.Button Compartment_DropDown matlab.ui.control.DropDown ShowcompartmentsetupButton matlab.ui.control.Button SampletimeLabel matlab.ui.control.Label

Samplingtime_Num_input matlab.ui.control.NumericEditField Volume_plot matlab.ui.control.UIAxes

VolumePanel matlab.ui.container.Panel V1Panel matlab.ui.container.Panel

V1Num_input matlab.ui.control.NumericEditField V1_Unit_DropDown matlab.ui.control.DropDown

V1Label matlab.ui.control.Label V2Panel matlab.ui.container.Panel

V4Label matlab.ui.control.Label FlowPanel matlab.ui.container.Panel Q12Panel matlab.ui.container.Panel

Q12Num_input matlab.ui.control.NumericEditField Q12_Unit_DropDown matlab.ui.control.DropDown

Q12Label matlab.ui.control.Label Q13Panel matlab.ui.container.Panel

FREDERICK HOLMER FANN JONAS STRÖM SEEZ

Development of an MRI-compatible

Multi-compartment Phantom for

Dynamic Studies