Cardiac Development Stages

Carnegie Stages are commonly used to outline discrete points during embryogenesis.

Define physical features of the embryo at specific times during its development

Cardiac development begins at Stage 9 and continues through to Stage 23

Carnegie Stage 9

Cardiac primordium forms

This cluster of cells eventually forms the heart.

Primary heart field, secondary heart field, myocardium, endocardium and neural tube

Carnegie Stages - Continued

Carnegie – Stage 23

By stage 23, the heart is structurally developed

Shunts still exist in the body, mostly bypassing the liver and lungs

These close at birth, restoring normal circulation

'Normal' Foetal Heart

At the end of normal development, the heart will have all major vessels and chambers.

Growth continues through the remainder of the pregnancy.

Heart – After the Birth

The foramen ovale closes, separating the atria

The ductus arteriosus closes, stopping the bypass of deoxygenated blood from the lungs.

Plan

Model development process for a normal heart, compare with a similar model for Tetralogy of Fallot.

Use differences between results from model to highlight the time periods during which structural differences begin occurring

Increase the detail of the model around these areas, repeating the comparisons at each stage.

Problems

Although development is well catalogued for the normal heart, there remains little definite information about the Tetralogy development

Specifics about gene involvement still unclear for either process, and there is an incredible amount of complexity behind morphogenesis

Existing information obtained from homologues such as mice and chicks, with different genetic information

Multiscale Modelling

There are a number of possible approaches to modelling a complex system.

Multiscale modelling is the process of creating a simulation to accurately represent the important features of a system at different spatial or temporal levels

The challenge lies in relating the different scales together to form a complete, working model

Multiscale Modelling Time Scales

Complete Pregnancy

Cardiac Development

Carnegie Stages

Cellular Growth and Differentiation (Mitosis)

Genetic and Chemical Signalling

Model will start spanning the entire cardiac development period and have details filled in at the shorter time-scales as needed

Multiscale Modelling – Time Domain

It is possible to break the cardiac development process into discrete stages

Using the Carnegie stages as a starting point it should be possible to model the processes behind the cardiac development and utilise said model for identifying potential causes for congenital heart defects.

Multiscale Modelling – Time Domain

Although not usually associated with biological process modelling, a finite state machine may be suitable for organising the processes driving cardiac development

As suggested by the name, FSMs are discrete models, with clearly defined conditions and pathways between them – they only have a finite number of states that they can be in at any point

Multiscale Modelling – Finite State Machine

Finite State Machines

Any FSM is comprised of three elements

States (reflects changes in the system from the initial conditions)

Transitions (shows the changes in states and defines the conditions for any change)

Actions (describes an activity to be undertaken)

Finite State Machines

All possible conditions must be defined else the model will fail to represent reality

All possible transitions must be made possible, which can lead to an immensely complex state machine

Discrete modelling process, no references to the actual time needed for a stage

Use of UML for State Machines

Expands on the traditional state machine architecture

Allows the use of hierarchical states within states, and extended global states which may be independent of the current location in the FSM

Processes can be orthogonalised if they have nothing in common, reducing the complexity of the transitions

Use of UML for State Machines

Example of a simple UML state machine

The boxes are states, the top half contains the actual state and the bottom half the processes within the state

UML State Machines

Extended States

Allows the usage of variables within the machine (for example a counter monitoring number of cycles performed)

Guard Conditions

Enable transitions only when specific criteria are met.

Needed for extended state usage, for example one option may become available only after 1000 steps but the actual steps don't matter

Hierarchically nestable states

Allows extremely complex systems to be built up at varying levels of detail

Therefore can create an general overview of a system, outlining the most general details, and use the ability to nest states to provide more detail if needed

UML State Machines

Orthogonal regions

If a particular state contains a number of sub-states which have no relationship it is possible to separate them.

This increases the simplicity of the final model as there are fewer needed transitions to model.

States are only truly orthogonal if there is no overlap at all. For example, it is possible to separate the number pad and the letter pad on a keyboard into orthogonal sections due to their separate special lock functions (“caps lock” forces capital letters, “num lock” enables the alternative functions and arrows on the number pad)

In the above example both sections belong to the “keyboard” state machine yet due to having no processes in common can be considered and evaluated separately

Standard State Machine

For comparison's sake, above is the same process shown using a standard Finite State Machine

There are an increased number of transitions for every stage to permit the global clear and off functions to be used

UML State Machine

Above is a UML-based state machine showing the effects of the two global commands (clear and power off)

Note how they are outside of the internal states and can be triggered at any time the calculator is in the 'on' state, regardless of what else is happening

State Machines for Biological Modelling?

Often, there are known stages in biological processes, with fairly well defined transitions between them.

In addition, these stages can often be broken down into having more detailed elements or can be considered to contain a number of sub-processes

One example that illustrates this rather well is the cell cycle

Cell Cycle Finite State Machine

At the most basic level, the cell cycle consists of two main stages, the Interphase (where individual cells grow and prepare for multiplication) and the Mitotic Phase (where the cell divides into two new cells)

Interphase Sub-State Machine However, each of the two

main stages can be considered to contain their own processes

Here, the interphase consists of two growth phases and a chromosome synthesis stage

The top level model would not be able to move on until the sub-level state machine reached its natural end

Mitosis Sub-State Machine

Similarly, the Mitotic Phase of the cell cycle can be broken down into sub-states which must occur in order before the cell cycle can proceed.

Each of the sub-states can be further broken down into more finely detailed state machines if required.

Using FSM for Cardiac Development

Finite state machines do not depend on a real timer, they are discrete systems whose states change depending on conditions

However, by controlling the available transitions between states and the processes during them it should be possible to model the process of cardiac development

Cardiac Development FSM

May be possible to use the Carnegie Stages as known states in a potential FSM-type model

Difficulties may occur when adding in detail at finer levels, as some processes may have cascading effects that are non-trivial to model.

In addition to this, not all of the effects of certain genes are known, nor are all of the processes controlling expression.