Modelling The Hemodynamics of Coronary Ischemia

Abstract

Acting upon clinical patient data, acquired in the pathway of percutaneous intervention, we deploy hierarchical, multi-stage, data-handling protocols and interacting low- and high-order mathematical models (chamber elastance, state-space system and CFD models), to establish and then validate a framework to quantify the burden of ischaemia. Our core tool is a compartmental, zero-dimensional model of the coupled circulation with four heart chambers, systemic and pulmonary circulations and an optimally adapted windkessel model of the coronary arteries that reflects the diastolic dominance of coronary flow. We guide the parallel development of protocols and models by appealing to foundational physiological principles of cardiac energetics and a parameterisation (stenotic Bernoulli resistance and micro-vascular resistance) of patients’ coronary flow. We validate our process first with results which substantiate our protocols and, second, we demonstrate good correspondence between model operation and patient data. We conclude that our core model is capable of representing (patho)physiological states and discuss how it can potentially be deployed, on clinical data, to provide a quantitative assessment of the impact, on the individual, of coronary artery disease.

1. Introduction

Myocardial function is dependent on sufficient coronary blood flow (CBF), the rate of which is matched closely to the fluctuating metabolic requirements of the heart. CBF is compromised in ischaemic heart disease (IHD), the commonest cause of death in the world. The most frequent cause of IHD is coronary artery disease (CAD), whereby the epicardial coronary arteries become stenosed or occluded, thus restricting CBF, initially under exercise conditions, but eventually at rest; see Section 3.2.2. Although there are data correlating global myocardial ischaemic burden with clinical outcomes, there are little data linking it with myocardial energetics, function or cardiac output. Here we demonstrate a multi-compartment, zero-dimensional (0D) model [1,2,3,4] of the coronary and systemic circulation, based on clinical data, with the potential to address myocardial energetics. Specifically, we postulate that a zero-dimensional (0D) model, adapted to readily available, patient-specific data, can accurately simulate coronary (patho)physiology, with the potential to personalise [5] and predict an individual patient’s global ischaemic burden and associate this with LV energetics and cardiac output. Our aims are to:

• Devise and implement a 0D model and robust data handling protocols able to replicate in silico haemodynamic dysfunction;
• Make optimal use of appropriate haemodynamic data, collected in the clinical PCI pathway;
• Demonstrate our process’ ability to capture patient-specific (patho)physiological states in silico.

2. Background

CFD modelling can provide a detailed assessment of local, usually single-vessel coronary haemodynamics which may be patient-specific [6]. It has also allowed for less or even non-invasive diagnostic tools, such as virtual fractional flow reserve (vFFR) [7], and in the novel form of the lattice Boltzmann simulation [8], it may be extended to describe flow within multiple small vessels [9]. However, CFD alone cannot provide a description of the perfusion of the entire myocardium necessary to characterise global ischaemic burden. To quantify the effects of multi-vessel disease upon total myocardial blood flow, one must set the diseased vessel(s) within the context of the whole coronary tree. This requires a methodology which can interact with CFD and wider physiological data.

Windkessel models are not new [10,11], neither are low-order representations of cardiac function [12,13,14] and coupled pseudo-mechanical system models, which have both been utilised here have been available, open source, for years [15]. We are concerned with computational medicine however, where a parsimonious model design should be constrained by data sources, with suitable interpretation protocols in mind. A composite of low-order models comprising a mechanical cardio-vascular 0D model of left heart function, valvular function and systemic circulation has been shown to have the capacity to ingest CFD-derived data and prognose the systemic effects of aortic and mitral valve disease [16]. Czechowicz et al. took CFD-derived valve properties, ingested them into a 0D model and formulated hypotheses connecting valve physiology, personalised cardiac energetics and valve function, to develop a clinical decision support tool. Here, we attempt to extend that methodology to a wider range of clinical data and disease states which are more difficult to characterise. This requires:

• A coronary network topology conforming to the clinical pattern of atherosclerosis;
• Modelling that reflects diastolic dominance of myocardial flow;
• The ability to incorporate all available clinical patient data;
• Sufficient computational simplicity for clinical deployment.

These demands make it necessary to simplify the task from the outset. To reduce the burden of model personalisation, we parameterise its right circulation parameters to normal values [17]. (This decision is supported by our input parameter sensitivity analysis in Appendix B, which shows that the coronary, left heart and systemic circulation input parameters are most influential). Further, the fact that resting coronary blood flow (250 mL min${}^{-1}$; 0.8 mL min${}^{-1}$ g${}^{-1}$ of heart muscle) constitutes about 5% of cardiac output [18] allows us to assume that coronary flow does not directly influence the systemic circulation. It also allows us to approach our problem by adjusting, with the relevant coronary flow data a posteriori, a model in which systemic and ventricular outputs are tuned first. Further, we appeal, hierarchically, to the key physiological principles of LV performance and the systemic circulation to construct each patient’s baseline and only then incorporate detailed CFD-derived information related to the coronaries’ microvascular resistance (MVR), etc., while retaining a minimum number of assumptions necessary to impose physiological principles. We use cardiac energetics and the concepts encapsulated in Wiggers’ diagram [19] to refine our low-order model of LV contractility and compliance; then, we consider ventricular–aortic (VA) coupling (which characterises the interaction between the myocardial contractile function and LV afterload and defines cardiovascular performance and efficiency [20]), and finally, the role of passive ventricular filling and valves. Only once an accurate model of general cardiac function is established will coronary flow be considered.

3. Methods and Materials

Broadly, we first describe our model (Section 3.1), then its data (Section 3.2). However, since model operation and data are related, some model details bleed into Section 3.2.

3.1. Methods

Personalisation cost function hyper-surfaces are intricate in the presence of non-linearities. Similar to valve functions, the cost function varies at very different rates with displacement in different parameter directions. Then, the task of locating the global minimum becomes computationally expensive. We address this difficulty with the multi-step protocol outlined in

Table 1. Stages in the personalisation process and their rationale. Physiological effects (column 2) are sequentially ingested into the workflow, with the objective declared in column 3, culminating, at step 5, in a patient-specific in silico model of the resting patient state.

Step	Processing Tool	Physiology	Objective	Comment
0	statistical analysis	LV function	ensemble average, downsample time-series	synchronised to ECG1 R-R interval
1	elastance model	LV function	derive LV elastance; ingest into 0D model	characterised by double Hill LV parameters
2	0D system model	LV–aortic coupling in systole	primary tune of 0D model systemic circulation windkessel parameters	large sub-space scanned by GA tuning
3	0D system model	LV–atrial coupling and diastolic filling	ancillary tuning of 0D left atrial elastance and mitral valve resistances	uses gradient descent tuning.
4	CFD analysis	coronary flows	ingest into 0D model computed $R_{sten}$, MVR	use software VirtuHEART${}^{\mathrm{TM}}$
5	0D system model	system-coronary	in silico representation of the patient at rest

. The six steps declared there buffer clinical data into our tool, using low-order haemodynamic models, a closed loop system model and finally 3D CFD.

3.1.1. Low-Order Models

To express heart chamber biomechanics and valve characteristics appropriately, we use the double Hill elastance model [12,14] and the valves’ pressure-flow characteristics. See Appendix A.2 for details; see also Czechowicz et al. [16], who give a fuller account of valve characteristic functions. Our passive systemic and pulmonary circulation windkessels, with proximal inertance (L), viscous dissipation (R) and multiple compliance (C), are also well documented [21]. Our pulmonary and systemic windkessels are designated LRCRC, having two capacitances to separate arterial and venous compliance. Note, each coronary branch is characterised by a compliance with its back plate pressure set to that in the relevant ventricle. Note also that elastance is not used solely to activate our system model. We will also use it, together with the LV PV loop [22], to clarify the relationship between our LV pressure and volume time-series data in Section 3.2.1.

3.1.2. System Model

Figure 1 shows, in relation to our CFD models, our assembly of windkessels, elastance functions and valves. It is designated a compartmental, electrical analogue or 0D model. It has four heart chambers as well as systemic arterial and venous, pulmonary arterial and venous, and several coronary arterial compartments. The overhead of four heart chambers is offset by enhanced performance [17]. We set the right circulation parameters to population average values. We compute N sampled time-series outputs $y_n\left(t_i\right)$, $n=1,..,N$, derived from internal states, $x_n\left(t_i\right)$, $n=1,..,N$ with the sample times $t_i$ drawn from the converged heart cycle.

Table 2. Indexing and naming conventions for system model variables. Free subscript * is to be replaced by the appropriate symbol. Flows generally are trans-compartment; pressures and volumes apply to compartments directly.

Subscript
Number	Name
	${\mathit{p}}_{*}\left(\mathit{t}\right),{\mathit{v}}_{*}\left(\mathit{t}\right)$	${\mathit{q}}_{*}\left(\mathit{t}\right)$
1	LV	ao val
2	RA	tri val
3	RV	pul val
4	LA	mit val
5	sys art	sys prox
6	pul art	pul prox
7	sys ven	sys dist
8	pul ven	pul dist
9	cor art	cor prox
10	cor ven	cor dist

declares our compartment numbering convention and

Table 3. System model input parameter nomenclature and status. The 0D model chamber parameters are classified by compartment. The status of the parameter is either fixed (at population average value) or tuned, from an initial value declared above. In the latter, the step in which the parameter is tuned is identified below. Note, the coronary LAD circulation parameters 74, …, 80 and the coronary RCA circulation parameters 81, …, 87 follow the naming convention of those declared above for the left circumflex artery, with the text “LAD” and “RCA” replacing the text “Cx” and identical initialisations.

Compartment	Index	Symbol	Description	Unit	Status	Base Value
NA	1	HR	Heart rate	bpm	Allocated
NA	2	mcfp	Mean circulatory filling pressure	mmHg	tuned, step 3	7
Left Ventricle (LV)	3	$E_{LV,min}$	LV min. elastance (“compliance”)	mmHg/mL	assigned, step2	0.08
	4	$E_{LV,max}$	LV max. elastance (“contractility”)	mmHg/mL	assigned, step3	3
	5	$n_{LV,1}$	LV contraction rate constant	-	tuned, step 2	1.32
	6	$n_{LV,2}$	LV relaxation rate constant	-	tuned, step 2	27.4
	7	${\tau }_{LV,1}$	LV systolic time constant	-	tuned, step 2	0.269
	8	${\tau }_{LV,2}$	LV diastolic time constant	-	tuned, step 2	0.452
	9	$V_{LV,0}$	LV volume offset	mL	fixed	10
	10	$T_{LV,shift}$	LV fractional time shift	-	fixed	0
Left Atrium (LA)	11	$E_{LA,min}$	LA max. elastance (“compliance”)	mmHg/mL	tuned, step 4	0.08
	12	$E_{LA,max}$	LA min. elastance (“contractility”)	mmHg/mL	tuned, step 4	0.17
	13	$n_{LA,1}$	LA contraction rate constant	-	fixed	1.32
	14	$n_{LA,2}$	LA relaxation time constant	-	fixed	13.1
	15	${\tau }_{LA,1}$	LA systolic time	-	tuned, step 4	0.11
	16	${\tau }_{LA,2}$	LA diastolic time	-	tuned, step4	0.18
	17	$V_{LA,0}$	LA volume offset	mL	fixed	3
	18	$T_{LA,shift}$	LA fractional time shift	-	fixed	0.85
Right Ventricle (RV)	19	$E_{RV,min}$	RV min. elastance (“compliance”)	mmHg/mL	fixed	0.04
	20	$E_{RV,max}$	RV max. elastance (“contractility”)	mmHg/mL	fixed	0.6
	21	$n_{RV,1}$	RV contraction rate constant	-	fixed	1.32
	22	$n_{RV,2}$	RV relaxation rate constant	-	fixed	27.4
	23	${\tau }_{RV,1}$	RV systolic time	-	fixed	0.269
	24	${\tau }_{RV,2}$	RV diastolic time	-	fixed	0.452
	25	$V_{RV,0}$	RV volume offset	mL	fixed	55
	26	$T_{RV,shift}$	RV fractional time shift	-l	fixed	0
Right Atrium (RA)	27	$E_{RA,min}$	RA max. elastance (“compliance”)	mmHg/mL	fixed	0.04
	28	$E_{RA,max}$	RA min. elastance (“contractility”)	mmHg/mL	fixed	0.15
	29	$n_{RA,1}$	RA contraction rate constant	-	fixed	1.32
	30	$n_{RA,2}$	RA relaxation rate constant	-	fixed	13.1
	31	${\tau }_{RA,1}$	RA systolic time	-	fixed	0.11
	32	${\tau }_{RA,2}$	RA diastolic time	-	fixed	0.18
	33	$V_{RA,0}$	RA volume offset	mL	fixed	17
	34	$T_{RA,shift}$	RA fractional time shift	-	fixed	0.85
Systemic Circulation LR CRC windkessel	35	$R_{sysart,dist}$	Systemic Resistance Distal	mmHg·s/mL	tuned, step 3	0.033
	36	$R_{sysart,prox}$	Systemic Resistance Proximal	mmHg·s/mL	tuned, step 3	0.9
	37	$C_{sysart,prox}$	Systemic Arterial Proximal Capacitance	mL/mmHg	tuned, step 3	1
	38	$V_{sysart,prox,0}$	Systemic Arterial Proximal Unstressed Volume	mL	fixed	720
	39	$C_{sysart,dist}$	Systemic Arterial Distal Capacitance	mL/mmHg	fixed	11
	40	$V_{sysart,dist,0}$	Systemic Arterial Distal Unstressed Volume	mL	fixed	3000
	41	$L_{sysart,prox}$	Systemic Arterial Inertance (Proximal)	mmHg·s2/mL	tuned, step 3	0.005
Pulmonary Circulation LR CRC windkessel	42	$R_{pulart,dist}$	Pulmonary Resistance Distal	mmHg·s/mL	fixed	0.01
	43	$R_{pulart,prox}$	Pulmonary Resistance Proximal	mmHg·s/mL	fixed	0.05
	44	$C_{pulart,prox}$	Pulmonary Arterial Proximal Capacitance	mL/mmHg	fixed	10
	45	$V_{pulart,prox,0}$	Pulmonary Arterial Proximal Unstressed Volume	mL	fixed	180
	46	$C_{pulart,dist}$	Pulmonary Arterial Distal Capacitance	mL/mmHg	fixed	15
	47	$V_{pulart,dist,0}$	Pulmonary Arterial Distal Unstressed Volume	mL	fixed	720
	48	$L_{pulart,prox}$	Pulmonary Arterial Inertance (Proximal)	mmHg·s2/mL	fixed	0.0017
Aortic Valve	49	$a_{2,aortic}$	Aortic Valve quadratic coefficient	mmHg·s${}^2$/mL${}^2$	fixed	1.60$\times {10}^{-5}$
	50	$a_{1,aortic}$	Aortic Valve linear coefficient	mmHg·s/mL	fixed	0
	51	$a_{2,aortic,regurg}$	Aortic Valve regurgitant quadratic coefficient	mmHg·s${}^2$/mL${}^2$	fixed	0
	52	$a_{1,aortic,regurg}$	Aortic Valve regurgitant linear coefficient	mmHg·s/mL	fixed	0
Mitral Valve	53	$a_{2,mitval}$	Mitral Valve quadratic coefficient	mmHg·s${}^2$/mL${}^2$	tuned, step 4	6.25$\times {10}^{-6}$
	54	$a_{1,mitval}$	Mitral Valve linear coefficient	mmHg·s/mL	tuned, step 4	0
	55	$a_{2,mitval,regurg}$	Mitral Valve regurgitant quadratic coefficient	mmHg·s${}^2$/mL${}^2$	fixed	0
	56	$a_{1,mitval,regurg}$	Mitral Valve regurgitant linear coefficient	mmHg·s/mL	fixed	0
Pulmonary Valve	57	$a_{2,pulval}$	Pulmonary Valve quadratic coefficient	mmHg·s${}^2$/mL${}^2$	fixed	8.16$\times {10}^{-6}$
	58	$a_{1,pulval}$	Pulmonary Valve linear coefficient	mmHg·s/mL	fixed	0
	59	$a_{2,pulval,regurg}$	Pulmonary Valve regurgitant quadratic coefficient	mmHg·s${}^2$/mL${}^2$	fixed	0
	60	$a_{1,pulval,regurg}$	Pulmonary Valve regurgitant linear coefficient	mmHg·s/mL	fixed	0
Tricuspid Valve	61	$a_{2,trival}$	Tricuspid Valve quadratic coefficient	mmHg·s${}^2$/mL${}^2$	fixed	6.25$\times {10}^{-6}$
	62	$a_{1,trival}$	Tricuspid Valve linear coefficient	mmHg·s/mL	fixed	0
	63	$a_{2,trival,regurg}$	Tricuspid Valve regurgitant quadratic coefficient	mmHg·s${}^2$/mL${}^2$	fixed	0
	64	$a_{1,trival,regurg}$	Tricuspid Valve regurgitant linear coefficient	mmHg·s/mL	fixed	0
Coronary Left Main	65	$a_{2,LMart,regurg}$	Left main stenosis, quadratic coefficient	mmHg·s${}^2$/mL${}^2$	assigned, step 5	0
	66	$a_{1,LMart,regurg}$	Left main stenosis, linear coefficient	mmHg·s/mL	assigned, step 5	0
Coronary Circumflex RCRC windkessel	67	$a_{2,Cx,art}$	Circumflex, quadratic coefficient	mmHg·s${}^2$/mL${}^2$	assigned, step 5	0
	68	$a_{1,Cx,art}$	Circumflex, linear coefficient	mmHg·s/mL	assigned, step 5	0
	69	$R_{Cx,MVR}$	Circumflex, RCR total resistance	mmHg·s/mL	assigned, step 5	35
	70	$C_{Cx}$	Circumflex, RCR capacitance	mL/mmHg	tuned, step 5	0.25
	71	${\alpha }_{Cx}$	Circumflex, RCR fraction proximal resistance	-	tuned, step 5	0.9
	72	$V_{Cx,0}$	Circumflex unstressed volume	mL	tuned, step 5	8
	73	${\beta }_{Cx}$	Circumflex capacitance back pressure fraction	-	fixed	0.5

our names, conventions, equivalent subscripting and input parameter base values. Note, micro-vascular resistance (MVR) is distributed relative to arterial compliance, with a proximal-to-distal ratio controlled by input parameter $\alpha$.

It is typical to express the dynamics of the system in Figure 1 in coupled ordinary differential equations (ODEs), or differential algebraic equations (DAEs [23]), typically with compartment pressures serving as internal states. In state-space form [24], using relative time, our model is expressed as

$$\frac{d\underline{x}}{d\tau }=\underline{f}(\underline{x}\left(\tau \right);\underline{\mathsf{\Theta }}),\underline{y}\left(\tau \right)=\underline{h}(\underline{x}\left(\tau \right);\underline{\mathsf{\Theta }}),$$

Vector function $\underline{h}(*)$, which maps between internal states and outputs, is often the identity. Outputs $\underline{y}\left(\tau \right)$ can, of course, be sampled to generate a discrete output metrics vector. $\underline{\mathsf{\Theta }}$ denotes the system model’s full input parameter set and

$$\underline{x}\left(\tau \right)={\left(p_1\left(\tau \right),p_2\left(\tau \right),..,p_{10}\left(\tau \right)\right)}^T.$$

is our instantaneous state vector. We use $\underline{\mathsf{\Theta }}$ ($\underline{\theta }$) to denote the full set (a subset) of input parameters. Particular examples of the system expressed in equations in Section 3.1.2 are given in Appendix A. Deriving equations in Section 3.1.2 uses recursive applications of simple haemodynamic sub-models, to formulate, in closed form, all the necessary relationships between compartmental pressures, volumes and flows (see Appendix A). A simple but lengthy process of eliminating flow and volume variables results in a DAE system [23] which is readily transformed into state-space form, by differentiating equations where necessary. Typically, the system is solved using a time-marching numerical solution, e.g., a fourth-order Runge–Kutta algorithm. A time step of $1\times {10}^{-3}$ was chosen for the numerical solution. It will occasionally be helpful to replace the numerical compartment index (subscript) with an abbreviation of its name, e.g., $p_1\left(t\right)=p_{LV}\left(t\right)$, with LV denoting left ventricle, etc.; see Table 2 and Table 3. For example model output data see Figure 2, which shows flow in a coronary artery illustrating a key feature- to capture diastolic dominance in coronary flow, our coronary vessels’ compliance is biased by the LV or RV pressure, as appropriate.

3.1.3. Input Parameters, Local Sensitivity and Orthogonality Analysis

Of our 87 input parameters, a majority take population average values and are not tuned; see Table 3. The interrogated input parameter sub-space is spanned by parameters of cardiac energetics, coronary perfusion, LV and left atrial elastance, the mitral valve, the coronary and systemic windkessels and mean circulatory filling pressure. The underlying full local sensitivity and input parameter orthogonality analysis is shown in Appendix B, along with a reduced analysis, based upon the most relevant inputs and outputs.

3.1.4. Model Personalisation or Input Parameter Identification

Personalisation involves inverse operation, guided by a cost function, together with an appropriate multi-variate functional minimisation algorithm. Our personalisation process is multi-stage. It is summarised in the inter-relating Table 1,

Table 4. Patient-specific data. Data are classified according to physical and mathematical significance, how the data are used, patient state and the 0D model compartment(s) to which they apply. Table 5 provides more information on the sub-models used to introduce these data into our 0D system model. The free subscript * in column 6 denotes one of the resolved coronary arteries, LMS (left main stem), LCA (left coronary artery), LCX (left circumflex) and RCA.

Data Source	Compartment	Role	Unit	Data Form	Notation	State			Pathway Stage
						Rest	Exercise	Hyperemia	Pre PCI	Post PCI
six-minute walk test	all (general)	cardiac output proxy	∼	statistics	∼	yes	yes	no	yes	yes
activity monitoring	all (general)	cardiac output proxy	∼	statistics	∼	yes	yes	no	yes	yes
CMR scan	left ventricle	volume	ml	time-series	$v_{LV}\left(t\right)$	yes	no	no	yes	no
coronary angiograms	coronaries	flow	mL/s	discrete	$<Q_{*}{>}_t$	yes	no	yes	yes	yes
coronary angiograms	coronaries	MVR	mmHg·s/mL	discrete	$R_{\mu LCA}$ etc.	yes	no	yes	yes	yes
PCI pressure catheter	left ventricle	pressure	mmHg	time-series	$p_{LV}\left(t\right)$	yes	no	yes	yes	yes
PCI pressure catheter	systemmic	pressure	mmHg	time-series	$p_{AO}\left(t\right)$	yes	no	yes	yes	no
PCI pressure catheter	coronaries	pressure	mmHg	time-series	$p_{*}\left(t\right)$ etc.	yes	no	yes	yes	yes

and

Table 5. Sub-models used to pre-process patient data. We provide further information on how patient data outlined in Table 4 are prepared. Here, $<.{>}_{EA}$ denotes an ensemble average, taken over equivalent intervals of dimensionless time $\tau$. This table shows the interaction between the various data sources and re-processing, to allow them to be ingested into the 0D compartmental system model (column 3). All pressure data were downsampled onto the CMR phase times; see Equation (2).

Source Data	Pre-Processing	Ingests into 0D System Model as
$<v_{LV}\left(\tau \right){>}_{EA}$	fit to double Hill elastance function	personal elastance function parameters
$<p_{LV}\left(\tau \right){>}_{EA}$
$<v_{LV}\left(\tau \right){>}_{EA}$	extract extrema	personal LV volume extrema
$<p_{LV}\left(\tau \right){>}_{EA}$	extract extrema	personal LV pressure extrema
$<p_{AO}\left(\tau \right){>}_{EA}$		personal systemmic (aortic) pressure time-series
coronary angiograms	steady-state CFD simulations	personal MVRs for treated vessels
$<p_{LMS}\left(\tau \right){>}_{EA}$ etc.
coronary angiograms	steady-state CFD simulation	personal stenotic Bernoulli resistance quadratic and linear coefficients in treated vessels
$<p_{LMS}\left(\tau \right){>}_{EA}$ etc.
6mwt	clinical interpretation	validation data
monitoring

. For all stages, the minimised cost function is a scalar weighted composite of: (i) the L${}^2$ norm of the difference between model-predicted and target data interpolated onto the same times, $t_i$, and (ii) a derived metric, $Y_n$

$$f\left(\underline{\mathsf{\Theta }}\right)=\sum _{j=1}^M\left(w_j^{\left(1\right)}\left\{\sum _{i=1}^N{\left(y_j\left(\underline{x}(t_i;\underline{\mathsf{\Theta }})\right)-y_j^{target}\left(t_i\right)\right)}^2\right\}+w_j^{\left(2\right)}{Y}_n^2\right).$$

(1)

For example, a derived metric might be systolic systemic arterial (compartment 5) pressure

$$Y_5=\underset{t_i}{max}\left(p_5(t_i;\underline{\mathsf{\Theta }})\right)-\underset{t}{max}\left(p_5^{target}\left(t\right)\right),$$

with $w_j^{\left(1\right)}$ and $w_j^{\left(2\right)}$ denoting the weights. We defer further discussion until Section 3.2.

Having summarised our processing and its rationale and declared our notation, we may now conclude this sub-section by defining the processing in key steps 3 and 4 of our protocol, which require inverse operation, or tuning, of the 0D system model. In

Table 6. Supplementary information for Table 1. Additional information on personalisation steps 3, 4, declared in Table 1. The search target data and input parameter sub-space are identified here. Note that search targets accumulate between steps 3 and 4.

Step	Personalisation Search Definition
	Input Parameters Adjusted	Output Metrics Targeted
3	mcfp, L, $R_{prox}$, $R_{dist}$, $C_{prox}$	${max}_{\tau }\left(v_{LV}\left(\tau \right)\right),{min}_{\tau }\left(v_{LV}\left(\tau \right)\right),{max}_{\tau }\left(p_{LV}\left(\tau \right)\right),{min}_{\tau }\left(p_{LV}\left(\tau \right)\right),\{p_{AO}\left({\tau }_i\right);i=1,..,29\}$
4	$a_{1,mit},a_{2,mit},n_1^{LA},n_2^{LA},{\tau }_1^{LA},{\tau }_2^{LA},E_{LA,max},E_{LA,min}$	to the above, add $\{v_{LV}\left({\tau }_i\right);i=1,..,29\}$

we state the model input parameters which are tuned and the corresponding patient data that is used to perform the tuning in each of the steps 3 and 4.

3.1.5. 3D CFD Models

We proceed to consider the CFD tools and modelling used in our study. Patient-specific stenotic Bernoulli resistance coefficients and MVRs are provided in our core model using two validated CFD based methods, (i) VirtuHEART${}^{\mathrm{TM}}$ (University of Sheffield, Sheffield, United Kingdom) and (ii) VirtuQ${}^{\mathrm{TM}}$ (University of Sheffield, Sheffield, United Kingdom) [6,7]. In the domain $\mathsf{\Omega }$, these tools solve the incompressible Navier–Stokes and continuity equations

$$\begin{array}{ccc}\hfill \left(\underline{v}\left(\underline{r}\right)·\underline{\nabla }\right)\underline{v}\left(\underline{r}\right)& =& -\frac{1}{\rho }\nabla p\left(\underline{r}\right)+\nu {\nabla }^2\underline{v}\left(\underline{r}\right),\hfill \\ \hfill \underline{\nabla }·\underline{v}\left(\underline{r}\right)& =& 0,\hfill \\ \hfill \end{array}$$

(2)

for steady, i.e., time-average flow; all symbols have their usual meaning. An accepted approximate value of Reynolds’ number $Re\approx 600$ for the coronary arteries, which indicates transitional flow. Note that we assume blood is a Newtonian fluid. Patient-specific boundary conditions are derived as follows. A closed coronary artery geometry (Figure 1d) is decomposed into a luminal boundary, $\partial {\mathsf{\Omega }}_1$, an entrance face, $\partial {\mathsf{\Omega }}_2$, and an exit face, $\partial {\mathsf{\Omega }}_3$. Using epipolar geometry and two compatible angiogram views, taken at the time of PCI, we construct $\partial {\mathsf{\Omega }}_1$, generate a mesh and apply a Dirichlet, no-slip condition

$$\underline{v}\left(\underline{r}\right)=0,\underline{r}\in \partial {\mathsf{\Omega }}_1.$$

We remark that a model of vessel sequestration using Murray’s law [25,26] to assign a flux across the luminal boundary based upon local geometry is available [7,27]; however, zero leak was specified here. Lesion proximal and distal time-averaged pressures are also recorded at PCI. Solutions were computed using Fluent${}^{\mathrm{TM}}$ (Ansys Corporation, Canonsburg, PA, United States of America), with boundaries $\partial {\mathsf{\Omega }}_2$ and $\partial {\mathsf{\Omega }}_3$ processed in each of two ways:

• Following VirtuQ methodology, apply measured proximal and distal average pressure on $\partial {\mathsf{\Omega }}_2$, $\partial {\mathsf{\Omega }}_3$, respectively,

  $$p\left(\underline{r}\right)={〈p_{*,prox}\left(\tau \right)〉}_{\tau },\underline{r}\in \partial {\mathsf{\Omega }}_2;p\left(\underline{r}\right)={〈p_{*,dist}\left(\tau \right)〉}_{\tau },\underline{r}\in \partial {\mathsf{\Omega }}_3,$$

  we compute the steady-state solution and deduce vessel flow

  $$q_{*}=\int {\int }_{\partial {\mathsf{\Omega }}_3}\underline{v}\left(\underline{r}\right)·d\underline{A};$$

  then, we use Ohm’s law and ${〈p_{*,dist}\left(\tau \right)〉}_{\tau }$ to compute a patient-specific MVR.
• Following VirtuHeart methodology, assume an MVR based on patient characteristics [6]; compute vessel inlet and outlet areas

  $$A_{in}=\int {\int }_{\partial {\mathsf{\Omega }}_2}dA,A_{out}=\int {\int }_{\partial {\mathsf{\Omega }}_3}dA,$$

  set two physiologically plausible inlet flows ($q_{*}=1,3$ mL/s), find uniform inlet and outlet streamwise velocities

  $$v_s^{in}\left(\underline{r}\right)=\frac{q_{*}}{A_{in}},\underline{r}\in \partial {\mathsf{\Omega }}_2;v_s^{out}\left(\underline{r}\right)=\frac{q_{*}}{A_{in}},\underline{r}\in \partial {\mathsf{\Omega }}_3,$$

  compute the two corresponding pressure drops, $\delta p_{*}^1$, $\delta p_{*}^3$ and hence infer the two Bernoulli resistance coefficients of the stenosis.

3.2. Materials

We analysed anonymised clinical datasets from 40 CAD patients with angina symptoms. The elective PCI workflow culminates in re-vascularisation of one or more of the coronaries in a majority of cases. The decision to stent is guided by the average pressure drop across a lesion. This is measured using a pressure wire (PRESSUREWIRE${}^{\mathrm{TM}}$X Abbott, downloading software Coroflow${}^{\mathrm{TM}}$, Coroventis Uppsala, Sweden) with the patient in a hyperaemic state of pharmacological stress induced by adenosine (a strong vasodilator); from this, one computes

$$FFR=\frac{p_{distal}}{p_{proximal}},$$

(3)

with re-vascularisation indicated for $FFR<0.8$. The clinical pathway generates PCI pressure data, cardiac MRI of the LV, six-minute walk tests (6mwt) before and after PCI and home activity monitoring (see below); these data fall into two categories

• Measured data (such as pressure and volume, expressed as time-series);
• Exercise and physical activity data (like patient 6mwt (exercise) and home monitoring (activity) measures).

Table 4 identifies data available for every patient, for Table 5, the particular form in which pre-processed data modalities, declared in Table 4, are ingested into the system model, and Table 1, a commensurate sequence of handling protocols.

Our personalisation process steps $0,\dots ,5$ (Table 1) introduce physiological effects sequentially, considering

• Functional form (qualitative data trends should accord with accepted physiology);
• Modality (pressure is measured directly, using a manometer; CMR volumes, in contradistinction, involve reciprocal space reconstruction).

Our raw data take the form of time-series at two sampling rates. Pressure data were all sampled for 120 s, using a sampling rate of 100 Hz. CMR volumes were acquired at a nominal sampling rate of 30 images per heart cycle. Pressure and volume can be co-registered to the start of ventricular contraction using the RR interval of the concurrent ECG1 traces, also recorded. Pressure and CMR data were acquired at different stages in the patient pathway (Table 4), with patients often in different physiological states. For all pressure signals, the sampled wave-train had its low frequency contribution removed and was decomposed, using time-domain analysis, into single beat pressure excursions, identified from the concurrent ECG 1 R-R interval, and then ectopic beats were filtered, and the remaining cycle pressure samples were downsampled onto the 30 dimensionless times defined as follows

$${\tau }_n=\frac{n}{30}\tau ,n\in \mathbb{N},0\le n\le 29;\tau =\frac{t}{T},T=\frac{60}{HR},$$

(4)

corresponding to CMR data sampling times. This set was then ensemble averaged, with the sample standard deviations in each dimensionless time channel providing an error. Example aortic and LV pressure data are shown in Figure 3. CMR-derived LV volume data are also effectively ensemble averaged (acquisition involves a reciprocal space reconstruction of data acquired at a given cycle phase, over several consecutive beats). Our CMR total LV volume measurements are attributed to 30 equispaced dimensionless times. Each was computed by segmenting the chamber, in a set of between 9 and 12 cross-sections, with the base and apical frames frequently discarded, owing to error. Frames were positioned along the long axis of the approximately cylindrical LV. Our segmentation and subsequent volume computation were performed using the MASS${}^{\mathrm{TM}}$ software (Leiden University, Version 2018 EXP, Leiden University Medical Centre, Leiden, The Netherlands). Figure 3, right panel, shows a typical CMR volume time-series.

Downsampled pressure and volume are acquired relative to a common clock provided by the ECG 1 RR interval which can potentially synchronise them, to yield a personal double Hill elastance function, $E_{LV}\left(t\right)$, etc. However, there are demonstrable inconsistencies, such as heart rate (HR) miss-match. Pressure data have a certain HR, and CMR data another. We will demonstrate that careful pressure and volume time-series co-registration is essential, and we will develop a minimal rigid shift which adjusts only relative phase.

3.2.1. Physical Data Processing

Chamber elastance functions (Appendix A.2) are low-order models developed from studied physical data. Elastance may be defined as follows

$$E_{*}\left(\tau \right)\equiv \frac{p_{*}\left(\tau \right)}{(v_{*}\left(\tau \right)-v_0)},$$

(5)

where subscript * now denotes the chamber and $v_0$ its unstressed volume. We take the clinical approximation, $v_0=0$. Clearly in Equation (5), $E_{*}\left(\tau \right)$ is sensitive to the relative phase of $p_{*}\left(\tau \right)$ and $v_{*}\left(\tau \right)$, but the importance of co-registration is perhaps better illustrated by another metric—the PV loop. The latter is a closed, two-dimensional parametric, Cartesian curve

$$(p_{LV}\left(\tau \right),v_{LV}\left(\tau \right)),\tau \in [0,1].$$

Figure 4, top panel, shows a personal PV loop in which the LV pressure and volumes are correctly co-registered. The ventricular–aortic coupling parameter (blue line) and the wasted mechanical work, as well as the useful mechanical work (loop area), can be deduced. In contradistinction, Figure 4, bottom panel, shows the same patient’s PV loop using the putative common clock of the RR interval, to co-register the two time-series signals (red data). This primitive co-registration is implausible: it has no identifiable iso-volumetric phases, and it implies a high level of valvular regurgitation and irregular aortic and mitral valve actuation pressures. The difference between Figure 4, top and bottom panels. argues for a principled co-registration other that the EGG 1 RR signal. Reserving for subsequent work a more elaborate co-registration, possibly involving signal dilations, we chose rigidly to delay the LV volume samples $\{v_{LV}\left({\tau }_n\right),0\le n\in \mathbb{N},n\le 29\}$ relative to a fixed pressure sample set $\{p_{LV}\left({\tau }_n\right),n\in \mathbb{N},n\le 29\}$, by an integral number of dimensionless time steps, $\delta \tau =({\tau }_n-{\tau }_{(n-1)})$. We observed a cohort-average shift of two CMR phases. Patient-specific synchronisation was accomplished by seeking the instant at which the aortic and mitral valves open, in the separate pressure and volume time-series, using the multiple criteria declared in the fourth column of

Table 7. Identification of mitral and aortic valve opening times, in LV pressure and volume time-series data. Each valve opening instant was identified by multiple criteria (column 3). Discrete time-series data temporal derivatives were evaluated by the class of finite difference approximation stated in column 4. Column 5 defines the notation for the identified cycle time fraction.

Valve	Time-Series	Valve Opening Signature	Difference	Notation
Aortic	pressure	$p_{LV}\gtrsim 80$ mmHg	central	$T_{ao}^p$
		${\max }_t\left(\frac{d{p}_{LV}}{dt}\right)$
	volume	$\frac{d{v}_{LV}}{dt}\lesssim 5$ mL/s	forward	$T_{ao}^v$
		close to ${\max }_t\left(v_{LV}\right)$
Mitral	pressure	$\frac{d{p}_{LV}}{dt}<0$	central	$T_{mit}^p$
		$\frac{d^2{p}_{LV}}{d{t}^2}\gtrsim 0$
		$p_{LV}\approx 10$ mmHg
	volume	$\left(\frac{d{v}_{LV}}{dt}\right)>0$	forward	$T_{mit}^v$
		$\left(\frac{d^2{v}_{LV}}{d{t}^2}\right)>0$

.

When times $T_{ao}^p$ and $T_{ao}^v$ are not simultaneous, a plausible co-registration is imposed by postulating a shift, or delay in the data $\{v_{LV}\left({\tau }_i\right),i=0,..,29\}$ of ${\delta }_t^{ao}=(T_{ao}^p-T_{ao}^v)$ CMR phases. Similarly, ${\delta }_t^{mit}=(T_{mit}^p-T_{mit}^v)$ was evaluated. The final rigid shift of CMR volume time-series data is

$$S\approx \frac{\left({\delta }_t^{ao}+{\delta }_t^{mit}\right)}{2},S\in \mathbb{N},0\le S\le 29.$$

Here, the symbol ≈ is used to denote a rounded value. The result of applying the rigid shift defined in Table 7 to our volume time-series data transforms the PV loop; see Figure 4, bottom panel. Our processing paradigm may be summarised as: constrain the LV pressure and volume time-series to conform to accepted PV loop physiology.

With an acceptable co-registration of pressure and volume data, we derive the LV double Hill elastance function parameters. $E_{LV,max}$ (widely termed “contractility”) and $E_{LV,min}$ (“compliance”) are assigned directly

$$E_{LV,max}=\underset{i}{max}\left(\frac{p_{LV}\left(t_i\right)}{v_{LV}\left(t_i\right)}\right),E_{LV,min}=\underset{i}{min}\left(\frac{p_{LV}\left(t_i\right)}{v_{LV}\left(t_i\right)}\right).$$

(6)

The remaining activation function parameters accrue by seeking the extremum of a multi-variate function, which is assumed to be differentiable

$$f(n_1,n_2,{\tau }_1,{\tau }_2)=\sum _{i=0}^{29}\left(p_{LV}\left({\tau }_i\right)-E({\tau }_i;\underline{\theta })v_{LV}\left({\tau }_i\right)\right),\underline{\theta }={\left(n_1,n_2,{\tau }_1,{\tau }_2\right)}^T\subset \underline{\mathsf{\Theta }}.$$

(7)

Note, $n_1,n_2,{\tau }_1,{\tau }_2$ are variables (parameters) on the left (right)-hand side of Equation (7). Minimisation was performed using a gradient descent, initialised to the population average [12]. Figure 5 below shows a derived elastance function.

In this way, our mechanical time-series signals are downsampled onto a common rate of $\frac{60}{30HR}$ Hz, suitably co-registered, and then used to deduce a personalised $E_{LV}\left(\tau \right)$ for each patient. These data will be recycled as 0D system model personalisation targets, both as discrete time-series and derived metrics.

3.2.2. Exercise/Monitoring Data Processing

CAD and cardiorespiratory fitness are linked. Endothelial nitric oxide synthase (eNOS) prevents platelet aggregation and white cell adhesion and inhibits vascular smooth muscle cell proliferation [28], and regular physical exercise enhances bioavailability of eNOS and regeneration of the vascular endothelium [29]. Conversely, inactivity is associated with pathological processes preceding atherosclerosis and CAD [30]. As a result, a dose–response relationship exists between physical activity and the risk of CAD [31], with exercise capacity predictive of mortality, myocardial infarction and risk of re-vascularisation in patients with established CAD [32]; see Figure 6. Assessing patients’ activity and exercise capacity is key in the management and assessment of CAD. For completeness, therefore, we summarise, in two categories, the study exercise and activity data which will, in future work, quantify the change in patient exercise tolerance attending measured PCI-related physiological changes.

• Home activity monitoring. Fitbit${}^{\mathrm{TM}}$ Charge 4 wrist watches (Healthy Metrics Research Inc. San Francisco, CA, USA) and a smartphone were used. Data monitoring started at recruitment and extended up to six months post PCI. All data were uploaded to the Fitbit website simultaneously through a smartphone application and then exported as coarse-grained time-series (mechanical energy consumption, distance walked, minutes spent sedentary, light, fairly, and great activity). Concurrent HR data were also exported as downsampled HR data, which were simultaneously updated throughout the day as time-averaged values every 10–15 min.
• Formal six-minute walk tests (6MWT). These were performed at baseline before the PCI (ideally one day) with a repeat assessment after three and six months. In these standard clinical assessments, the data collected were cuff blood pressure, HR and distance walked.

We remark that exercise training in CAD is superior to PCI in improving event-free survival and exercise capacity and is at a much lower cost [33], and the total distance walked is a proxy for cumulative CO.

4. Results

Three randomly chosen patients designated A, …, C were used to validate our personalisation process, by demonstrating an accurate in silico description of patients’ pre-PCI rest state. Since changes in arterial geometry accompanying re-vascularisation are recorded, PCI-modified MVR and stenotic Bernoulli resistance coefficients will accrue with our approach. Thus, data such as those presented here will eventually be available on the post-operative patient; see Section 5. Results fall into two categories: (i) data on our methodology as summarised in Table 1 and Table 4 and (ii) the emergent patient representations.

4.1. Elastance Function Evaluation

Downsampling and ensemble averaging of patient LV and aortic pressure time-series datasets were performed straightforwardly. We proceed to protocol step 2 (a personal $E_{LV}\left(\tau \right)$)), where we encounter the sensitivity of $E_{LV}\left(\tau \right)$, to the co-registration of pressure and volume data. See

Table 8. Sensitivity of LV double Hill elastance function parameterisation. For a single patient B, fitted double Hill elastance function shape parameters are declared for a range of co-registration shifts. We deem a shift of 5 in the last column to be the correct co-registration for this particular patient. The corresponding elastance function sequence is plotted in Figure 7.

	Shift	1	2	3	4	5
double Hill LV elastance parameter	$n_1$	1.91	1.83	1.78	1.68	1.52
	$n_2$	16.46	17.93	18.06	17.52	17.04
	${\tau }_1$	0.28	0.32	0.38	0.47	0.73
	${\tau }_2$	0.47	0.48	0.49	0.49	0.49
	$E_{LV,min}$	0.04	0.04	0.04	0.04	0.04
	$E_{LV,max}$	2.95	2.86	2.68	2.48	2.27

and Figure 7, which describe patient B, who required the largest shift. The PV loop illustrates the importance of co-registration, in Figure 4, where the PV loop shape progressively acquires accepted physiological features of iso-volumetric compression and relaxation phases. The corresponding sensitivity of $E_{LV}\left(\tau \right)$, in Figure 7, is quantified by the parameterisations in Table 8. We defer further comment to Section 5.

Figure 8 shows the personalised $E_{LV}\left(\tau \right)$ (left) and PV loop (right) for patients A, …, C. The elastance data take correctly co-registered LV pressure and volumes and show the initial (red) and optimised (black) fit. The PV loops in the right-hand column show the importance of co-registration case by case, with the PV loop obtained without co-registration shown in red and that with an appropriate co-registration in black.

Figure 7. Sensitivity of a patient elastance function to co-registration shift. Data correspond to the double Hill elastance function parameters in Table 8. The shift increases top to bottom with the bottom panel corresponding to the expected shift of 5 phases, deduced using the co-registration declared in Table 9. The broken line shows our common initialisation using Mynard’s population average [12] parameterisation. Open red circles show the raw data with the appropriate shift applied. The black circles and the line show the optimised fit on the shifted data from a gradient descent simplex method applied to Equation (7).

Figure 7. Sensitivity of a patient elastance function to co-registration shift. Data correspond to the double Hill elastance function parameters in Table 8. The shift increases top to bottom with the bottom panel corresponding to the expected shift of 5 phases, deduced using the co-registration declared in Table 9. The broken line shows our common initialisation using Mynard’s population average [12] parameterisation. Open red circles show the raw data with the appropriate shift applied. The black circles and the line show the optimised fit on the shifted data from a gradient descent simplex method applied to Equation (7).

Table 9. Patient-specific LV double Hill elastance function parameters. These parameterisations result from fitting the product $v_{LV}\left({\tau }_i\right)\times E({\tau }_i;\underline{\theta })$ to the measured, downsampled pressure $p_{LV}\left({\tau }_i\right)$, using the cost function defined in Equation (7) and a gradient descent method. The applied co-registration rigid shift, or relative delay, applied to the volume data is declared in the first row.

Patient ID		A	B	C
Co-Registration Rigid Shift		1	5	2
double Hill LV elastance parameter	${\tau }_1$	0.29	0.54	0.82
	${\tau }_2$	0.42	0.5	0.42
	$n_1$	1.58	1.56	1.26
	$n_2$	17.53	17.96	19.95
	$E_{LV,max}$	1.1	2.22	2.17
	$E_{LV,min}$	0.05	0.03	0.03

Table 9. Patient-specific LV double Hill elastance function parameters. These parameterisations result from fitting the product $v_{LV}\left({\tau }_i\right)\times E({\tau }_i;\underline{\theta })$ to the measured, downsampled pressure $p_{LV}\left({\tau }_i\right)$, using the cost function defined in Equation (7) and a gradient descent method. The applied co-registration rigid shift, or relative delay, applied to the volume data is declared in the first row.

Patient ID		A	B	C
Co-Registration Rigid Shift		1	5	2
double Hill LV elastance parameter	${\tau }_1$	0.29	0.54	0.82
	${\tau }_2$	0.42	0.5	0.42
	$n_1$	1.58	1.56	1.26
	$n_2$	17.53	17.96	19.95
	$E_{LV,max}$	1.1	2.22	2.17
	$E_{LV,min}$	0.05	0.03	0.03

4.2. Tuning Protocol Evaluation

Steps 0, 1, and 2 culminate in a personal $E_{LV}\left(\tau \right)$ which is input into the system 0D model. The latter then provides time-series outputs for LV and aortic pressures and LV volume at the end of step 3; see Figure 9. The cost function used to acquire these data, Equation (1), equally weights its five targets. Step 3 uses a genetic algorithm (GA) to localise the global cost function minimum, by adjusting the systemic windkessel. In general, the LV diastolic phase exhibits the largest discrepancy between model outputs and targets. This suggests parameters, such as mitral valve resistances and left atrial (LA) elastance, as secondary targets for step 4, which uses a less expensive gradient descent search, on the assumption that previous GA tuning will first have located the correct region. The initial values of the input parameters of the mitral valve Bernoulli resistance and $E_{LA}\left(\tau \right)$ are specified in Table 1 and Table 5. Step 4 retains the targets of step 3 but adds the LV volume time-series. That step 4 favours the diastolic phase, particularly for cases A and C, is apparent from Figure 9, which shows outputs at the end of step 3 in the left column and outputs from the end of step 4 in the right column.

Having assigned systemic and left-heart energetic parameters, it remains to tune flow in patients’ coronary circulations. The coronaries sequester a small fraction of cardiac output [18]. It is reasonable to suppose that this does not perturb already identified systemic and left heart parameters. The tuning of coronaries in step 5 relies upon CFD data to assign the MVR and stenotic Bernoulli resistance coefficients. The latter rely on data declared in

Table 10. Coronary artery average pressure data. Time average pressure for lesion-proximal, aortic sinus, and lesion-distal locations. Study data are recorded for treated arteries only. Pressures were obtained using a pressure transducer by catheterisation during the PCI process.

Patient	Vessel
	LAD		LCx		RCA
	$p_{\mathit{prox}}$	$p_{\mathit{dist}}$	$p_{\mathit{prox}}$	$p_{\mathit{dist}}$	$p_{\mathit{prox}}$	$p_{\mathit{dist}}$
UNIT	mmHg	mmHg	mmHg	mmHg	mmHg	mmHg
A	88.60	74.50	∼	∼	∼	∼
B	103.71	99.17	103.95	101.69	∼	∼
C	121.74	115.73	∼	∼	127.23	125.89

and

Table 11. CFD-derived data for the patient coronary arteries. Stenotic Bernoulli resistance coefficients and MVRs for patients A,B,C.

Patient	Metric	Unit	Vessel
			LAD	LCx	RCA
A	MVR	mmHg·s/mL	1.50	∼	∼
	$R_{sten,a}$	mmHg·s${}^2$/mL${}^2$	28.45	∼	∼
	$R_{sten,b}$	mmHg·s/mL	28.58	∼	∼
B	MVR	mmHg·s/mL	0.77	1.10	∼
	$R_{sten,a}$	mmHg·s${}^2$/mL${}^2$	16.56	10.22	∼
	$R_{sten,b}$	mmHg·s/mL	22.34	14.33	∼
C	MVR	mmHg·s/mL	1.75	∼	1.01
	$R_{sten,a}$	mmHg·s${}^2$/mL${}^2$	22.34	∼	16.67
	$R_{sten,b}$	mmHg·s/mL	12.33	∼	14.11

and are derived as described in Section 4.3.

4.3. CFD Data Evaluation

Patients’ stenotic Bernoulli resistance coefficients were derived, as discussed above, from VIRTUheart${}^{\mathrm{TM}}$ (University of Sheffield, Sheffield, UK). VirtuQ${}^{\mathrm{TM}}$ (University of Sheffield, Sheffield, UK) was used to compute the corresponding MVRs. Both tools assume steady flow. Furthermore, VIRTUheart approximates pressure distal to the MVR as zero. The first assumption is raised within our transient 0D system model. Moreover, the 0D system model, with its four hear chambers, develops an unconstrained pressure distal to the MVR. The stenotic Bernoulli resistance coefficients ingested into the system 0D model (Table 10) are deemed to be unaffected by these differences. However, the MVR yielded by our CFD tool requires the correction in Equation (8)

$$MVR\to \left(\frac{{〈p_d\left(\tau \right)〉}_{\tau }-{〈p_{RA}\left(\tau \right)〉}_{\tau }}{{〈p_d\left(\tau \right)〉}_{\tau }}\right)MVR.$$

(8)

Table 10 and Table 11 declare the patient-specific CFD data inputted into the system 0D model. There, we encounter that sparsity and incompleteness which are typical of clinical data- none of our selected patients have three vessel disease and others have total coronary occlusion (TCO). The latte precludes all treatment but which can still be mapped to the coronary artery module in Figure 1 however, by setting large stenotic resistance coefficients. The latter are used to determine the pressure drop, $\delta P_{*}$, across a lesion from the flow, using Equation (9)

$$\delta P_{*}=R_{sten*,a}{q}_{*}^2+R_{sten*,b}{q}_{*}.$$

(9)

Above, * identifies the particular vessel.

5. Discussion

We have characterised the pre-PCI patient state, but equivalent data exist for post-PCI patient states; thus, the approach described here may be used to examine post-intervention physiology. Our methodology brings within scope for the first time, a means to relate qualitative changes in physiology to changes in, e.g., exercise tolerance. The importance of a careful co-registration of our LV pressure and volume time-series data is underscored by the data in Figure 7 and the corresponding double Hill parameterisations, declared in Table 8. These data show LV contractility ($E_{LV,max}$) and systolic activation function parameters ${\tau }_1$, $n_1$ to be the most sensitive elastance parameters to co-registration shift. These parameters respectively quantify LV contractility and the vigour of systolic onset and are very important indicators of cardiac health.

Step 2 of our protocol is seen to be robust and necessary. It represents a successful means of dealing with nonsimultaneous pressure and volume time-series data. From the data of Figure 8, it is apparent that a range of elastance functions and PV loops are successfully addressed. The change in the elastance function as correct co-registration is progressively applied demonstrates a significant degree of sensitivity in the elastance function. This underscores the importance of step 2 in our protocol. From Figure 8, it is also apparent that the PV loop of some cases is rather more sensitive than others to our rigid shift co-registration of pressure and volume time-series data.

From Figure 8, we see that the optimisation process enshrined in Equation (7) makes relatively little difference to the final shape of the elastance function, certainly compared with that produced by the volume and pressure time-series co-registrations.

6. Conclusions

We have verified the ability of appropriate protocols, PCI-acquired data and a suitable 0D system model, robustly interacting with 3D CFD models to recover a quantitative in silico representation of three individual patients’ physiological and pre-PCI cardiovascular state. This depiction relies on parsing diverse clinical data, compiled during the elective PCI clinical pathway. Specifically, our hierarchical, multi-stage protocol is shown to identify prioritised input model parameters of cardiac energetics and coronary perfusion, within a 0D (compartmental) four-chamber system model with data-adapted coronary artery topology. Using this, we reached verifiable in silico representations of significant LV aortic and coronary artery volume, flow and pressure time-series. This process of input parameter identification is known in the clinical context as model personalisation. It is a key determinant of clinical utility, which urges a parsimonious model or (as here) the prioritisation of a subset of model input parameters, chosen with the supporting sensitivity and orthogonality analysis, provided in the appendices. The present study requires accurate representation of coronary flow states which are dominated by diastolic dynamics. The coupling of the coronary arteries’ compliances (capacitances) to the appropriate chamber pressure (see Figure 1) apparently recovers diastolic dominance in coronary flow.

Time-series data play a defining role. To utilise them optimally, it was necessary to reconcile data collected at different stages of the clinical pathway with patients at different points in their physiological envelopes. This is deemed to be our largest source of error. The time-series data in question are non-invasive left ventricular volume, derived from CMR, and invasive LV and aortic pressures measured during PCI. Our approach involves downsampling pressure to accord with MRI phases and devising a suitable rigid phase shift, based upon LV valve timings, to overcome an identified co-registration error; results were assessed by the qualitative credibility of a physiological properties inferred from the LV PV loop. The commensurate LV elastance function is an input into the system 0D model personalisation process. It is shown to be sensitive to the co-registration of time-series data. Possibly, the most consequential, purely clinical outcome of this work is that we have been able to obtain credible results for patient PV loops and left-ventricular elastance functions by using a straightforward co-registration of non-simultaneous invasive pressure and non-invasive volume data when guided by rudimentary and therefore robust physiological principles.

Despite sources of error, it is apparent that there remains sufficient depth in the clinical pathway dataset to build appropriate digital twins, and we present as evidence data for three patients in the pre-PCI state. A route to the complementary, post-intervention description of the patient now lies open. One simply needs to apply the current workflow to extant data which are of equivalent form to that considered here. Therefore, using our methodology, one can expect to develop both pre- and post-PCI representation of patients from which to measure physiological changes attending PCI. Section 3.2.2 outlines a range of relevant activity measures which might be used meaningfully to contextualise this change in global ischaemic burden. Put another way, the act of correlating these model-derived statistics of the study data has the potential to reveal relationships between coronary physiology, cardiac energetics and physical activity, as well as providing clinical decision support. The model represents a means not only to establish a connection between global coronary perfusion and cardiac energetics, but to quantify it.